Multiplication and division of numbers with powers. Rule for multiplying powers with different bases
I. Product of powers with the same bases.
The product of two powers with the same bases can always be represented as a power with base x.
By definition, the power x 7 is the product of seven factors, each of which is equal to x, and x 9 is the product of nine of the same factors. Therefore, x 7 x 9 is equal to the product of 7 + 9 factors. Each of which is equal to x, that is
x 7 x 9 = x 7+9 = x 16
It turns out that if the base of degree a is an arbitrary number, and m and n are any natural numbers, then the equality is true:
a m · a n = a m + n
This equality expresses one of the properties of degree.
The product of two powers with the same base is equal to a power with the same base and an exponent equal to the sum of the exponents of these powers.
This property also occurs in cases where the number of factors is more than two.
For example, in the case of three factors we have:
a m · a n · a k = (a m · a n)a k = a m+n · a k = a m+n+k
When performing transformations, it is convenient to use the rule: when multiplying powers with the same bases, the bases are left the same, and the exponents are added.
Let's look at examples.
Example 1.
x 6 x 5 = x 6+5 = x 11
Example 2.
a 7 a -8 = a -1
Example 3.
6 1.7 6 - 0.9 = 6 1.7+(- 0.9) = 6 1.7 - 0.9 = 6 0.8
II. Partials of degrees with the same bases.
The quotient of two powers with the same exponents can always be represented as a power with the same base.
Let's look at examples.
Example 1. The quotient x 17: x 5 can be represented as a power with a base x:
x 17: x 5 = x 12,
since by definition of the quotient and based on the property of the degree x 5 · x 12 = x 17. The exponent of the quotient (number 12) is equal to the difference between the exponents of the dividend and the divisor (17 – 5):
x 17: x 5 = x 17-5
Example 2.
8 16: 8 12 = 8 16-12 = 8 4
Example 3.
a -8: a 6 = a -8-6 = a -14
Example 4.
b 5: b -4 = b 5-(-4) = b 9
Example 5
9 1.5: 9 - 0.5 = 9 1.5 - (- 0.5) = 9 1.5 + 0.5 = 9 2
When performing transformations, it is convenient to use the rule: when dividing powers with the same bases, the bases are left the same, and the exponent of the divisor is subtracted from the exponent of the dividend.
Example 6.
a 4: a 4 = a 4-4 = a 0
The value of the expression a 0 for any a ≠ 0 is equal to 1.
III. Raising a degree to a degree.
Let the seventh power of the expression a 2 be represented as a power with base a.
By definition, the power (a 2) 7 is the product of seven factors, each of which is equal to a 2, that is
(a 2) 7 = a 2 · a 2 · a 2 × a 2 · a 2 · a 2 · a 2 .
Applying the power property, we get:
a 2 · a 2 · a 2 · a 2 · a 2 · a 2 · a 2 = a 2+2+2+2+2+2+2 = a 2·7 .
It turns out, (a 2) 7 = a 2 7 = a 14.
When raising a power to a power, the base is left the same, and the exponents are multiplied:
(a m) n = a mn .
Let's look at examples.
Example 1.
(4 3) 4 = 4 3 4 = 4 12
Example 2.
((-2) 2) 5 = (-2) 10 = 1024
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After the power of a number has been determined, it is logical to talk about degree properties. In this article we will give the basic properties of the power of a number, while touching on all possible exponents. Here we will provide proofs of all properties of degrees, and also show how these properties are used when solving examples.
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Properties of degrees with natural exponents
By definition of a power with a natural exponent, the power a n is the product of n factors, each of which is equal to a. Based on this definition, and also using properties of multiplication of real numbers, we can obtain and justify the following properties of degree with natural exponent:
- the main property of the degree a m ·a n =a m+n, its generalization;
- property of quotient powers with identical bases a m:a n =a m−n ;
- product power property (a·b) n =a n ·b n , its extension;
- property of the quotient to the natural degree (a:b) n =a n:b n ;
- raising a degree to a power (a m) n =a m·n, its generalization (((a n 1) n 2) …) n k =a n 1 ·n 2 ·…·n k;
- comparison of degree with zero:
- if a>0, then a n>0 for any natural number n;
- if a=0, then a n =0;
- if a<0 и показатель степени является четным числом 2·m , то a 2·m >0 if a<0 и показатель степени есть нечетное число 2·m−1 , то a 2·m−1 <0 ;
- if a and b are positive numbers and a
- if m and n are natural numbers such that m>n , then at 0
- if m and n are natural numbers such that m>n , then at 0
Let us immediately note that all written equalities are identical subject to the specified conditions, both their right and left parts can be swapped. For example, the main property of the fraction a m ·a n =a m+n with simplifying expressions often used in the form a m+n =a m ·a n .
Now let's look at each of them in detail.
Let's start with the property of the product of two powers with the same bases, which is called the main property of the degree: for any real number a and any natural numbers m and n, the equality a m ·a n =a m+n is true.
Let us prove the main property of the degree. By the definition of a power with a natural exponent, the product of powers with the same bases of the form a m ·a n can be written as a product. Due to the properties of multiplication, the resulting expression can be written as 
, and this product is a power of the number a with a natural exponent m+n, that is, a m+n. This completes the proof.
Let us give an example confirming the main property of the degree. Let's take degrees with the same bases 2 and natural powers 2 and 3, using the basic property of degrees we can write the equality 2 2 ·2 3 =2 2+3 =2 5. Let's check its validity by calculating the values of the expressions 2 2 · 2 3 and 2 5 . Performing exponentiation, we have 2 2 ·2 3 =(2·2)·(2·2·2)=4·8=32 and 2 5 =2·2·2·2·2=32, since equal values are obtained, then the equality 2 2 ·2 3 =2 5 is correct, and it confirms the main property of the degree.
The basic property of a degree, based on the properties of multiplication, can be generalized to the product of three or more powers with the same bases and natural exponents. So for any number k of natural numbers n 1, n 2, …, n k the following equality is true: a n 1 ·a n 2 ·…·a n k =a n 1 +n 2 +…+n k.
For example, (2,1) 3 ·(2,1) 3 ·(2,1) 4 ·(2,1) 7 = (2,1) 3+3+4+7 =(2,1) 17 .
We can move on to the next property of powers with a natural exponent – property of quotient powers with the same bases: for any non-zero real number a and arbitrary natural numbers m and n satisfying the condition m>n, the equality a m:a n =a m−n is true.
Before giving the proof of this property, let us discuss the meaning of the additional conditions in the formulation. The condition a≠0 is necessary in order to avoid division by zero, since 0 n =0, and when we got acquainted with division, we agreed that it is impossible to divide by zero. The condition m>n is introduced so that we do not go beyond natural exponents. Indeed, for m>n the exponent a m−n is a natural number, otherwise it will be either zero (which happens for m−n ) or a negative number (which happens for m Proof. The main property of a fraction allows us to write the equality a m−n ·a n =a (m−n)+n =a m. From the obtained equality a m−n ·a n =a m and from it follows that a m−n is a quotient of powers of a m and a n . This proves the property of quotient powers with identical bases. Let's give an example. Let's take two degrees with the same bases π and natural exponents 5 and 2, the considered property of the degree corresponds to the equality π 5: π 2 = π 5−3 = π 3. Now let's consider product power property: the natural power n of the product of any two real numbers a and b is equal to the product of the powers a n and b n , that is, (a·b) n =a n ·b n . Indeed, by the definition of a degree with a natural exponent we have Here's an example: This property extends to the power of the product of three or more factors. That is, the natural power property n of the product of k factors is written as (a 1 ·a 2 ·…·a k) n =a 1 n ·a 2 n ·…·a k n. For clarity, we will show this property with an example. For the product of three factors to the power of 7 we have . The following property is property of a quotient in kind: the quotient of real numbers a and b, b≠0 to the natural power n is equal to the quotient of powers a n and b n, that is, (a:b) n =a n:b n. The proof can be carried out using the previous property. So (a:b) n b n =((a:b) b) n =a n, and from the equality (a:b) n ·b n =a n it follows that (a:b) n is the quotient of a n divided by b n . Let's write this property using specific numbers as an example: Now let's voice it property of raising a power to a power: for any real number a and any natural numbers m and n, the power of a m to the power of n is equal to the power of the number a with exponent m·n, that is, (a m) n =a m·n. For example, (5 2) 3 =5 2·3 =5 6. The proof of the power-to-degree property is the following chain of equalities: The property considered can be extended to degree to degree to degree, etc. For example, for any natural numbers p, q, r and s, the equality It remains to dwell on the properties of comparing degrees with a natural exponent. Let's start by proving the property of comparing zero and power with a natural exponent. First, let's prove that a n >0 for any a>0. The product of two positive numbers is a positive number, as follows from the definition of multiplication. This fact and the properties of multiplication suggest that the result of multiplying any number of positive numbers will also be a positive number. And the power of a number a with natural exponent n, by definition, is the product of n factors, each of which is equal to a. These arguments allow us to assert that for any positive base a, the degree a n is a positive number. Due to the proven property 3 5 >0, (0.00201) 2 >0 and It is quite obvious that for any natural number n with a=0 the degree of a n is zero. Indeed, 0 n =0·0·…·0=0 . For example, 0 3 =0 and 0 762 =0. Let's move on to negative bases of degree. Let's start with the case when the exponent is an even number, let's denote it as 2·m, where m is a natural number. Then Finally, when the base a is a negative number and the exponent is an odd number 2 m−1, then Let's move on to the property of comparing powers with the same natural exponents, which has the following formulation: of two powers with the same natural exponents, n is less than the one whose base is smaller, and greater is the one whose base is larger. Let's prove it. Inequality a n properties of inequalities a provable inequality of the form a n is also true (2.2) 7 and It remains to prove the last of the listed properties of powers with natural exponents. Let's formulate it. Of the two degrees with natural indicators and the same positive bases, less than one, the degree is greater, the indicator of which is less; and of two degrees with natural indicators and the same bases greater than one, the degree is greater, the indicator of which is greater. Let us proceed to the proof of this property. Let us prove that for m>n and 0 It remains to prove the second part of the property. Let us prove that for m>n and a>1 a m >a n is true. The difference a m −a n after taking a n out of brackets takes the form a n ·(a m−n −1) . This product is positive, since for a>1 the degree a n is a positive number, and the difference a m−n −1 is a positive number, since m−n>0 due to the initial condition, and for a>1 the degree a m−n is greater than one . Consequently, a m −a n >0 and a m >a n , which is what needed to be proven. This property is illustrated by the inequality 3 7 >3 2.
. Based on the properties of multiplication, the last product can be rewritten as 
, which is equal to a n · b n .
.
.
.
. For greater clarity, here is an example with specific numbers: (((5,2) 3) 2) 5 =(5,2) 3+2+5 =(5,2) 10
.
.
. For each of the products of the form a·a is equal to the product of the moduli of the numbers a and a, which means it is a positive number. Therefore, the product will also be positive 
and degree a 2·m. Let's give examples: (−6) 4 >0 , (−2,2) 12 >0 and .
. All products a·a are positive numbers, the product of these positive numbers is also positive, and its multiplication by the remaining negative number a results in a negative number. Due to this property (−5) 3<0
, (−0,003) 17 <0
и 
.
.
Properties of powers with integer exponents
Since positive integers are natural numbers, then all the properties of powers with positive integer exponents coincide exactly with the properties of powers with natural exponents listed and proven in the previous paragraph.
We defined a degree with an integer negative exponent, as well as a degree with a zero exponent, in such a way that all properties of degrees with natural exponents, expressed by equalities, remained valid. Therefore, all these properties are valid for both zero exponents and negative exponents, while, of course, the bases of the powers are different from zero.
So, for any real and non-zero numbers a and b, as well as any integers m and n, the following are true: properties of powers with integer exponents:
- a m ·a n =a m+n ;
- a m:a n =a m−n ;
- (a·b) n =a n ·b n ;
- (a:b) n =a n:b n ;
- (a m) n =a m·n ;
- if n is a positive integer, a and b are positive numbers, and a b-n;
- if m and n are integers, and m>n , then at 0
When a=0, the powers a m and a n make sense only when both m and n are positive integers, that is, natural numbers. Thus, the properties just written are also valid for the cases when a=0 and the numbers m and n are positive integers.
It is not difficult to prove each of these properties, for this it is enough to use the definitions of the degree with a natural and integer exponent, as well as the properties of actions with real numbers. As an example, let's prove that the power property holds for both positive integers and nonpositive integers. To do this, you need to show that if p is zero or a natural number and q is zero or a natural number, then the equalities (a p) q =a p·q, (a −p) q =a (−p)·q, (a p ) −q =a p·(−q) and (a −p) −q =a (−p)·(−q). Let's do it.
For positive p and q, the equality (a p) q =a p·q was proven in the previous paragraph. If p=0, then we have (a 0) q =1 q =1 and a 0·q =a 0 =1, whence (a 0) q =a 0·q. Similarly, if q=0, then (a p) 0 =1 and a p·0 =a 0 =1, whence (a p) 0 =a p·0. If both p=0 and q=0, then (a 0) 0 =1 0 =1 and a 0·0 =a 0 =1, whence (a 0) 0 =a 0·0.
Now we prove that (a −p) q =a (−p)·q . By definition of a power with a negative integer exponent, then 
. By the property of quotients to powers we have 
. Since 1 p =1·1·…·1=1 and , then . The last expression is, by definition, a power of the form a −(p q) , which, by virtue of the multiplication rules, can be written as a (−p) q .
Likewise 
.
AND 
.
By the same principle, one can prove all other properties of a degree with an integer exponent, written in the form of equalities.
In the penultimate of the recorded properties, it is worth dwelling on the proof of the inequality a −n >b −n, which is valid for any negative integer −n and any positive a and bfor which the condition a is satisfied . Since by condition a 0 . The product a n · b n is also positive as the product of positive numbers a n and b n . Then the resulting fraction is positive as the quotient of the positive numbers b n −a n and a n ·b n . Therefore, whence a −n >b −n , which is what needed to be proved.
The last property of powers with integer exponents is proven in the same way as a similar property of powers with natural exponents.
Properties of powers with rational exponents
We defined a degree with a fractional exponent by extending the properties of a degree with an integer exponent to it. In other words, powers with fractional exponents have the same properties as powers with integer exponents. Namely:
The proof of the properties of degrees with fractional exponents is based on the definition of a degree with a fractional exponent, and on the properties of a degree with an integer exponent. Let us provide evidence.
By definition of a power with a fractional exponent and , then 
. The properties of the arithmetic root allow us to write the following equalities. Further, using the property of a degree with an integer exponent, we obtain , from which, by the definition of a degree with a fractional exponent, we have 
, and the indicator of the degree obtained can be transformed as follows: . This completes the proof.
The second property of powers with fractional exponents is proved in an absolutely similar way: 
The remaining equalities are proved using similar principles: 
Let's move on to proving the next property. Let us prove that for any positive a and b, a b p . Let's write the rational number p as m/n, where m is an integer and n is a natural number. Conditions p<0 и p>0 in this case will be equivalent to the conditions m<0 и m>0 accordingly. For m>0 and a
Similarly, for m<0 имеем a m >b m , whence , that is, and a p >b p .
It remains to prove the last of the listed properties. Let us prove that for rational numbers p and q , p>q for 0 
And 
. And the definition of a degree with a rational exponent allows us to move on to inequalities and, accordingly. From this we draw the final conclusion: for p>q and 0
Properties of powers with irrational exponents
From the way a degree with an irrational exponent is defined, we can conclude that it has all the properties of degrees with rational exponents. So for any a>0 , b>0 and irrational numbers p and q the following are true properties of powers with irrational exponents:
- a p ·a q =a p+q ;
- a p:a q =a p−q ;
- (a·b) p =a p ·b p ;
- (a:b) p =a p:b p ;
- (a p) q =a p·q ;
- for any positive numbers a and b , a 0 the inequality a p b p ;
- for irrational numbers p and q , p>q at 0
From this we can conclude that powers with any real exponents p and q for a>0 have the same properties.
Bibliography.
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics Zh textbook for 5 cells. educational institutions.
- Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 7th grade. educational institutions.
- Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8th grade. educational institutions.
- Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 9th grade. educational institutions.
- Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the beginnings of analysis: Textbook for grades 10 - 11 of general education institutions.
- Gusev V.A., Mordkovich A.G. Mathematics (a manual for those entering technical schools).
Addition and subtraction of powers
It is obvious that numbers with powers can be added like other quantities , by adding them one after another with their signs.
So, the sum of a 3 and b 2 is a 3 + b 2.
The sum of a 3 - b n and h 5 -d 4 is a 3 - b n + h 5 - d 4.
Odds the same powers of the same variables can be added or subtracted.
So, the sum of 2a 2 and 3a 2 is equal to 5a 2.
It is also obvious that if you take two squares a, or three squares a, or five squares a.
But degrees various variables And various degrees identical variables, must be composed by adding them with their signs.
So, the sum of a 2 and a 3 is the sum of a 2 + a 3.
It is obvious that the square of a, and the cube of a, is not equal to twice the square of a, but to twice the cube of a.
The sum of a 3 b n and 3a 5 b 6 is a 3 b n + 3a 5 b 6.
Subtraction powers are carried out in the same way as addition, except that the signs of the subtrahends must be changed accordingly.
Or:
2a 4 - (-6a 4) = 8a 4
3h 2 b 6 - 4h 2 b 6 \u003d -h 2 b 6
5(a - h) 6 - 2(a - h) 6 = 3(a - h) 6
Multiplying powers
Numbers with powers can be multiplied, like other quantities, by writing them one after the other, with or without a multiplication sign between them.
Thus, the result of multiplying a 3 by b 2 is a 3 b 2 or aaabb.
Or:
x -3 ⋅ a m = a m x -3
3a 6 y 2 ⋅ (-2x) = -6a 6 xy 2
a 2 b 3 y 2 ⋅ a 3 b 2 y = a 2 b 3 y 2 a 3 b 2 y
The result in the last example can be ordered by adding identical variables.
The expression will take the form: a 5 b 5 y 3.
By comparing several numbers (variables) with powers, we can see that if any two of them are multiplied, then the result is a number (variable) with a power equal to amount degrees of terms.
So, a 2 .a 3 = aa.aaa = aaaaa = a 5 .
Here 5 is the power of the multiplication result, which is equal to 2 + 3, the sum of the powers of the terms.
So, a n .a m = a m+n .
For a n , a is taken as a factor as many times as the power of n;
And a m is taken as a factor as many times as the degree m is equal to;
That's why, powers with the same bases can be multiplied by adding the exponents of the powers.
So, a 2 .a 6 = a 2+6 = a 8 . And x 3 .x 2 .x = x 3+2+1 = x 6 .
Or:
4a n ⋅ 2a n = 8a 2n
b 2 y 3 ⋅ b 4 y = b 6 y 4
(b + h - y) n ⋅ (b + h - y) = (b + h - y) n+1
Multiply (x 3 + x 2 y + xy 2 + y 3) ⋅ (x - y).
Answer: x 4 - y 4.
Multiply (x 3 + x – 5) ⋅ (2x 3 + x + 1).
This rule is also true for numbers whose exponents are negative.
1. So, a -2 .a -3 = a -5 . This can be written as (1/aa).(1/aaa) = 1/aaaaa.
2. y -n .y -m = y -n-m .
3. a -n .a m = a m-n .
If a + b are multiplied by a - b, the result will be a 2 - b 2: that is
The result of multiplying the sum or difference of two numbers is equal to the sum or difference of their squares.
If you multiply the sum and difference of two numbers raised to square, the result will be equal to the sum or difference of these numbers in fourth degrees.
So, (a - y).(a + y) = a 2 - y 2.
(a 2 - y 2)⋅(a 2 + y 2) = a 4 - y 4.
(a 4 - y 4)⋅(a 4 + y 4) = a 8 - y 8.
Division of degrees
Numbers with powers can be divided like other numbers, by subtracting from the dividend, or by placing them in fraction form.
Thus, a 3 b 2 divided by b 2 is equal to a 3.
Writing a 5 divided by a 3 looks like $\frac $. But this is equal to a 2 . In a series of numbers
a +4 , a +3 , a +2 , a +1 , a 0 , a -1 , a -2 , a -3 , a -4 .
any number can be divided by another, and the exponent will be equal to difference indicators of divisible numbers.
When dividing degrees with the same base, their exponents are subtracted..
So, y 3:y 2 = y 3-2 = y 1. That is, $\frac = y$.
And a n+1:a = a n+1-1 = a n . That is, $\frac = a^n$.
Or:
y 2m: y m = y m
8a n+m: 4a m = 2a n
12(b + y) n: 3(b + y) 3 = 4(b + y) n-3
The rule is also true for numbers with negative degree values.
The result of dividing a -5 by a -3 is a -2.
Also, $\frac: \frac = \frac .\frac = \frac = \frac $.
h 2:h -1 = h 2+1 = h 3 or $h^2:\frac = h^2.\frac = h^3$
It is necessary to master multiplication and division of powers very well, since such operations are very widely used in algebra.
Examples of solving examples with fractions containing numbers with powers
1. Decrease the exponents by $\frac $ Answer: $\frac $.
2. Decrease exponents by $\frac$. Answer: $\frac$ or 2x.
3. Reduce the exponents a 2 /a 3 and a -3 /a -4 and bring to a common denominator.
a 2 .a -4 is a -2 the first numerator.
a 3 .a -3 is a 0 = 1, the second numerator.
a 3 .a -4 is a -1 , the common numerator.
After simplification: a -2 /a -1 and 1/a -1 .
4. Reduce the exponents 2a 4 /5a 3 and 2 /a 4 and bring to a common denominator.
Answer: 2a 3 /5a 7 and 5a 5 /5a 7 or 2a 3 /5a 2 and 5/5a 2.
5. Multiply (a 3 + b)/b 4 by (a - b)/3.
6. Multiply (a 5 + 1)/x 2 by (b 2 - 1)/(x + a).
7. Multiply b 4 /a -2 by h -3 /x and a n /y -3 .
8. Divide a 4 /y 3 by a 3 /y 2 . Answer: a/y.
Properties of degree
We remind you that in this lesson we will understand properties of degrees with natural indicators and zero. Powers with rational exponents and their properties will be discussed in lessons for 8th grade.
A power with a natural exponent has several important properties that allow us to simplify calculations in examples with powers.
Property No. 1
Product of powers
When multiplying powers with the same bases, the base remains unchanged, and the exponents of the powers are added.
a m · a n = a m + n, where “a” is any number, and “m”, “n” are any natural numbers.
This property of powers also applies to the product of three or more powers.
- Simplify the expression.
b b 2 b 3 b 4 b 5 = b 1 + 2 + 3 + 4 + 5 = b 15 - Present it as a degree.
6 15 36 = 6 15 6 2 = 6 15 6 2 = 6 17 - Present it as a degree.
(0.8) 3 · (0.8) 12 = (0.8) 3 + 12 = (0.8) 15 - Write the quotient as a power
(2b) 5: (2b) 3 = (2b) 5 − 3 = (2b) 2 - Calculate.
Please note that in the specified property we were talking only about the multiplication of powers with the same bases. It does not apply to their addition.
You cannot replace the sum (3 3 + 3 2) with 3 5. This is understandable if
calculate (3 3 + 3 2) = (27 + 9) = 36, and 3 5 = 243
Property No. 2
Partial degrees
When dividing powers with the same bases, the base remains unchanged, and the exponent of the divisor is subtracted from the exponent of the dividend.
11 3 − 2 4 2 − 1 = 11 4 = 44
Example. Solve the equation. We use the property of quotient powers.
3 8: t = 3 4
Answer: t = 3 4 = 81
Using properties No. 1 and No. 2, you can easily simplify expressions and perform calculations.
Example. Simplify the expression.
4 5m + 6 4 m + 2: 4 4m + 3 = 4 5m + 6 + m + 2: 4 4m + 3 = 4 6m + 8 − 4m − 3 = 4 2m + 5
Example. Find the value of an expression using the properties of exponents.
2 11 − 5 = 2 6 = 64
Please note that property 2 dealt only with the division of powers with the same bases.
You cannot replace the difference (4 3 −4 2) with 4 1. This is understandable if you calculate (4 3 −4 2) = (64 − 16) = 48, and 4 1 = 4
Property No. 3
Raising a degree to a power
When raising a power to a power, the base of the power remains unchanged, and the exponents are multiplied.
(a n) m = a n · m, where “a” is any number, and “m”, “n” are any natural numbers.
We remind you that a quotient can be represented as a fraction. Therefore, we will dwell on the topic of raising a fraction to a power in more detail on the next page.
How to multiply powers
How to multiply powers? Which powers can be multiplied and which cannot? How to multiply a number by a power?
In algebra, you can find a product of powers in two cases:
1) if the degrees have the same bases;
2) if the degrees have the same indicators.
When multiplying powers with the same bases, the base must be left the same, and the exponents must be added:
When multiplying degrees with the same indicators, the overall indicator can be taken out of brackets:
Let's look at how to multiply powers using specific examples.
The unit is not written in the exponent, but when multiplying powers, they take into account:
When multiplying, there can be any number of powers. It should be remembered that you don’t have to write the multiplication sign before the letter:
In expressions, exponentiation is done first.
If you need to multiply a number by a power, you should first perform the exponentiation, and only then the multiplication:
Multiplying powers with the same bases
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In this lesson we will study multiplication of powers with like bases. First, let us recall the definition of degree and formulate a theorem on the validity of the equality . Then we will give examples of its application on specific numbers and prove it. We will also apply the theorem to solve various problems.
Topic: Power with a natural exponent and its properties
Lesson: Multiplying powers with the same bases (formula)
1. Basic definitions
Basic definitions:
n- exponent,
— n th power of a number.
2. Statement of Theorem 1
Theorem 1. For any number A and any natural n And k the equality is true:
In other words: if A– any number; n And k natural numbers, then:
Hence rule 1:
3. Explanatory tasks
Conclusion: special cases confirmed the correctness of Theorem No. 1. Let us prove it in the general case, that is, for any A and any natural n And k.
4. Proof of Theorem 1
Given a number A– any; numbers n And k – natural. Prove:
The proof is based on the definition of degree.
5. Solving examples using Theorem 1
Example 1: Think of it as a degree.
To solve the following examples, we will use Theorem 1.
and) 
6. Generalization of Theorem 1
A generalization used here:
7. Solving examples using a generalization of Theorem 1
8. Solving various problems using Theorem 1
Example 2: Calculate (you can use the table of basic degrees).
A) 
(according to the table)
b) 
Example 3: Write as a power with base 2.
A) 
Example 4: Determine the sign of the number:
, A - negative because the exponent at -13 is odd.
Example 5: Replace ( ) with a power with a base r:
We have, that is.
9. Summing up
1. Dorofeev G.V., Suvorova S.B., Bunimovich E.A. and others. Algebra 7. 6th edition. M.: Enlightenment. 2010
1. School Assistant (Source).
1. Express as a degree:
a B C D E) 
3. Write as a power with base 2:
4. Determine the sign of the number:
A) 
5. Replace ( ) with a power of a number with a base r:
a) r 4 · (·) = r 15; b) (·) · r 5 = r 6
Multiplication and division of powers with the same exponents
In this lesson, we will study the multiplication of powers with the same exponents. First, let's recall the basic definitions and theorems about multiplying and dividing powers with the same bases and raising powers to powers. Then we formulate and prove theorems on multiplication and division of powers with the same exponents. And then with their help we will solve a number of typical problems.
Reminder of basic definitions and theorems
Here a- the basis of the degree,
— n th power of a number.
Theorem 1. For any number A and any natural n And k the equality is true:
When multiplying powers with the same bases, the exponents are added, the base remains unchanged.
Theorem 2. For any number A and any natural n And k, such that n > k the equality is true:
When dividing degrees with the same bases, the exponents are subtracted, but the base remains unchanged.
Theorem 3. For any number A and any natural n And k the equality is true:
All the above theorems were about powers with the same reasons, this lesson will consider degrees with the same indicators.
Examples for multiplying powers with the same exponents
Consider the following examples:
Let's write down the expressions for determining the degree.
Conclusion: From the examples it can be seen that 
, but this still needs to be proven. Let us formulate the theorem and prove it in the general case, that is, for any A And b and any natural n.
Formulation and proof of Theorem 4
For any numbers A And b and any natural n the equality is true:
Proof Theorem 4 .
By definition of degree:
So we have proven that .
To multiply powers with the same exponents, it is enough to multiply the bases and leave the exponent unchanged.
Formulation and proof of Theorem 5
Let us formulate a theorem for dividing powers with the same exponents.
For any number A And b() and any natural n the equality is true:
Proof Theorem 5 .
Let's write down the definition of degree:
Statement of theorems in words
So, we have proven that .
To divide powers with the same exponents into each other, it is enough to divide one base by another, and leave the exponent unchanged.
Solving typical problems using Theorem 4
Example 1: Present as a product of powers.
To solve the following examples, we will use Theorem 4.
To solve the following example, recall the formulas:
Generalization of Theorem 4
Generalization of Theorem 4:
Solving Examples Using Generalized Theorem 4
Continuing to solve typical problems
Example 2: Write it as a power of the product.
Example 3: Write it as a power with exponent 2.
Calculation examples
Example 4: Calculate in the most rational way.
2. Merzlyak A.G., Polonsky V.B., Yakir M.S. Algebra 7. M.: VENTANA-GRAF
3. Kolyagin Yu.M., Tkacheva M.V., Fedorova N.E. and others. Algebra 7.M.: Enlightenment. 2006
2. School assistant (Source).
1. Present as a product of powers:
A) ; b) ; V) ; G) ;
2. Write as a power of the product:
3. Write as a power with exponent 2:
4. Calculate in the most rational way.
Mathematics lesson on the topic “Multiplication and division of powers”
Sections: Mathematics
Pedagogical goal:
Tasks:
Activity units of teaching: determination of degree with a natural indicator; degree components; definition of private; combinational law of multiplication.
I. Organizing a demonstration of students’ mastery of existing knowledge. (step 1)
a) Updating knowledge:
2) Formulate a definition of degree with a natural exponent.
a n =a a a a … a (n times)
b k =b b b b a… b (k times) Justify the answer.
II. Organization of self-assessment of the student’s degree of proficiency in current experience. (step 2)
Self-test: (individual work in two versions.)
A1) Present the product 7 7 7 7 x x x as a power:
A2) Represent the power (-3) 3 x 2 as a product
A3) Calculate: -2 3 2 + 4 5 3
I select the number of tasks in the test in accordance with the preparation of the class level.
I give you the key to the test for self-test. Criteria: pass - no pass.
III. Educational and practical task (step 3) + step 4. (the students themselves will formulate the properties)
While solving problems 1) and 2), students propose a solution, and I, as a teacher, organize the class to find a way to simplify powers when multiplying with the same bases.
Teacher: come up with a way to simplify powers when multiplying with the same bases.
An entry appears on the cluster:
The topic of the lesson is formulated. Multiplication of powers.
Teacher: come up with a rule for dividing powers with the same bases.
Reasoning: what action is used to check division? a 5: a 3 = ? that a 2 a 3 = a 5
I return to the diagram - a cluster and add to the entry - .. when dividing, we subtract and add the topic of the lesson. ...and division of degrees.
IV. Communicating to students the limits of knowledge (as a minimum and as a maximum).
Teacher: the minimum task for today’s lesson is to learn to apply the properties of multiplication and division of powers with the same bases, and the maximum task is to apply multiplication and division together.
We write on the board : a m a n = a m+n ; a m: a n = a m-n
V. Organization of studying new material. (step 5)
a) According to the textbook: No. 403 (a, c, e) tasks with different wordings
No. 404 (a, d, f) independent work, then I organize a mutual check, give the keys.
b) For what value of m is the equality valid? a 16 a m = a 32; x h x 14 = x 28; x 8 (*) = x 14
Assignment: come up with similar examples for division.
c) No. 417 (a), No. 418 (a) Traps for students: x 3 x n = x 3n; 3 4 3 2 = 9 6 ; a 16: a 8 = a 2.
VI. Summarizing what has been learned, conducting diagnostic work (which encourages students, and not the teacher, to study this topic) (step 6)
Diagnostic work.
Test(place the keys on the back of the dough).
Task options: represent the quotient x 15 as a power: x 3; represent as a power the product (-4) 2 (-4) 5 (-4) 7 ; for which m is the equality a 16 a m = a 32 valid? find the value of the expression h 0: h 2 at h = 0.2; calculate the value of the expression (5 2 5 0) : 5 2 .
Lesson summary. Reflection. I divide the class into two groups.
Find arguments in group I: in favor of knowing the properties of the degree, and group II - arguments that will say that you can do without properties. We listen to all the answers and draw conclusions. In subsequent lessons, you can offer statistical data and call the rubric “It’s beyond belief!”
VII. Homework.
Historical reference. What numbers are called Fermat numbers.
P.19. #403, #408, #417
Used Books:
Properties of degrees, formulations, proofs, examples.
After the power of a number has been determined, it is logical to talk about degree properties. In this article we will give the basic properties of the power of a number, while touching on all possible exponents. Here we will provide proofs of all properties of degrees, and also show how these properties are used when solving examples.
Page navigation.
Properties of degrees with natural exponents
By definition of a power with a natural exponent, the power a n is the product of n factors, each of which is equal to a. Based on this definition, and also using properties of multiplication of real numbers, we can obtain and justify the following properties of degree with natural exponent:
- if a>0, then a n>0 for any natural number n;
- if a=0, then a n =0;
- if a 2·m >0 , if a 2·m−1 n ;
- if m and n are natural numbers such that m>n, then for 0m n, and for a>0 the inequality a m >a n is true.
- a m ·a n =a m+n ;
- a m:a n =a m−n ;
- (a·b) n =a n ·b n ;
- (a:b) n =a n:b n ;
- (a m) n =a m·n ;
- if n is a positive integer, a and b are positive numbers, and a n n and a −n >b −n ;
- if m and n are integers, and m>n , then for 0m n , and for a>1, the inequality a m >a n is satisfied.
Let us immediately note that all written equalities are identical subject to the specified conditions, both their right and left parts can be swapped. For example, the main property of the fraction a m ·a n =a m+n with simplifying expressions often used in the form a m+n =a m ·a n .
Now let's look at each of them in detail.
Let's start with the property of the product of two powers with the same bases, which is called the main property of the degree: for any real number a and any natural numbers m and n, the equality a m ·a n =a m+n is true.
Let us prove the main property of the degree. By the definition of a power with a natural exponent, the product of powers with identical bases of the form a m ·a n can be written as the product 
. Due to the properties of multiplication, the resulting expression can be written as 
, and this product is a power of the number a with a natural exponent m+n, that is, a m+n. This completes the proof.
Let us give an example confirming the main property of the degree. Let's take degrees with the same bases 2 and natural powers 2 and 3, using the basic property of degrees we can write the equality 2 2 ·2 3 =2 2+3 =2 5. Let's check its validity by calculating the values of the expressions 2 2 · 2 3 and 2 5 . Carrying out exponentiation, we have 2 2 2 3 =(2 2) (2 2 2) = 4 8 = 32 and 2 5 =2 2 2 2 2 = 32 , since we get equal values, then the equality 2 2 ·2 3 =2 5 is correct, and it confirms the main property of the degree.
The basic property of a degree, based on the properties of multiplication, can be generalized to the product of three or more powers with the same bases and natural exponents. So for any number k of natural numbers n 1 , n 2 , …, n k the equality a n 1 ·a n 2 ·…·a n k =a n 1 +n 2 +…+n k is true.
For example, (2,1) 3 ·(2,1) 3 ·(2,1) 4 ·(2,1) 7 = (2,1) 3+3+4+7 =(2,1) 17.
We can move on to the next property of powers with a natural exponent – property of quotient powers with the same bases: for any non-zero real number a and arbitrary natural numbers m and n satisfying the condition m>n, the equality a m:a n =a m−n is true.
Before giving the proof of this property, let us discuss the meaning of the additional conditions in the formulation. The condition a≠0 is necessary in order to avoid division by zero, since 0 n =0, and when we got acquainted with division, we agreed that we cannot divide by zero. The condition m>n is introduced so that we do not go beyond the natural exponents. Indeed, for m>n the exponent a m−n is a natural number, otherwise it will be either zero (which happens for m−n) or a negative number (which happens for m m−n ·a n =a (m−n) +n =a m. From the resulting equality a m−n ·a n =a m and from the connection between multiplication and division it follows that a m−n is a quotient of powers a m and an n. This proves the property of quotients of powers with the same bases.
Let's give an example. Let's take two degrees with the same bases π and natural exponents 5 and 2, the equality π 5:π 2 =π 5−3 =π 3 corresponds to the considered property of the degree.
Now let's consider product power property: the natural power n of the product of any two real numbers a and b is equal to the product of the powers a n and b n , that is, (a·b) n =a n ·b n .
Indeed, by the definition of a degree with a natural exponent we have 
. Based on the properties of multiplication, the last product can be rewritten as 
, which is equal to a n · b n .
Here's an example: 
.
This property extends to the power of the product of three or more factors. That is, the property of natural degree n of a product of k factors is written as (a 1 ·a 2 ·…·a k) n =a 1 n ·a 2 n ·…·a k n .
For clarity, we will show this property with an example. For the product of three factors to the power of 7 we have .
The following property is property of a quotient in kind: the quotient of real numbers a and b, b≠0 to the natural power n is equal to the quotient of powers a n and b n, that is, (a:b) n =a n:b n.
The proof can be carried out using the previous property. So (a:b) n ·b n =((a:b)·b) n =a n , and from the equality (a:b) n ·b n =a n it follows that (a:b) n is the quotient of division a n to b n .
Let's write this property using specific numbers as an example: 
.
Now let's voice it property of raising a power to a power: for any real number a and any natural numbers m and n, the power of a m to the power of n is equal to the power of the number a with exponent m·n, that is, (a m) n =a m·n.
For example, (5 2) 3 =5 2·3 =5 6.
The proof of the power-to-degree property is the following chain of equalities: .
The property considered can be extended to degree to degree to degree, etc. For example, for any natural numbers p, q, r and s, the equality 
. For greater clarity, let's give an example with specific numbers: (((5,2) 3) 2) 5 =(5,2) 3+2+5 =(5,2) 10.
It remains to dwell on the properties of comparing degrees with a natural exponent.
Let's start by proving the property of comparing zero and power with a natural exponent.
First, let's prove that a n >0 for any a>0.
The product of two positive numbers is a positive number, as follows from the definition of multiplication. This fact and the properties of multiplication suggest that the result of multiplying any number of positive numbers will also be a positive number. And the power of a number a with natural exponent n, by definition, is the product of n factors, each of which is equal to a. These arguments allow us to assert that for any positive base a, the degree a n is a positive number. Due to the proven property 3 5 >0, (0.00201) 2 >0 and 
.
It is quite obvious that for any natural number n with a=0 the degree of a n is zero. Indeed, 0 n =0·0·…·0=0 . For example, 0 3 =0 and 0 762 =0.
Let's move on to negative bases of degree.
Let's start with the case when the exponent is an even number, let's denote it as 2·m, where m is a natural number. Then . According to the rule for multiplying negative numbers, each of the products of the form a·a is equal to the product of the absolute values of the numbers a and a, which means that it is a positive number. Therefore, the product will also be positive 
and degree a 2·m. Let's give examples: (−6) 4 >0 , (−2,2) 12 >0 and .
Finally, when the base a is a negative number and the exponent is an odd number 2 m−1, then . All products a·a are positive numbers, the product of these positive numbers is also positive, and its multiplication by the remaining negative number a results in a negative number. Due to this property (−5) 3 17 n n is the product of the left and right sides of n true inequalities a properties of inequalities, a provable inequality of the form a n n is also true. For example, due to this property, the inequalities 3 7 7 and 
.
It remains to prove the last of the listed properties of powers with natural exponents. Let's formulate it. Of the two degrees with natural indicators and the same positive bases, less than one, the degree is greater, the indicator of which is less; and of two degrees with natural indicators and the same bases greater than one, the degree is greater, the indicator of which is greater. Let us proceed to the proof of this property.
Let us prove that for m>n and 0m n . To do this, we write down the difference a m − a n and compare it with zero. The recorded difference, after taking a n out of brackets, will take the form a n ·(a m−n−1) . The resulting product is negative as the product of a positive number a n and a negative number a m−n −1 (a n is positive as the natural power of a positive number, and the difference a m−n −1 is negative, since m−n>0 due to the initial condition m>n, whence it follows that when 0m−n is less than unity). Therefore, a m −a n m n , which is what needed to be proven. As an example, we give the correct inequality.
It remains to prove the second part of the property. Let us prove that for m>n and a>1 a m >a n is true. The difference a m −a n after taking a n out of brackets takes the form a n ·(a m−n −1) . This product is positive, since for a>1 the degree a n is a positive number, and the difference a m−n −1 is a positive number, since m−n>0 due to the initial condition, and for a>1 the degree a m−n is greater than one . Consequently, a m −a n >0 and a m >a n , which is what needed to be proven. This property is illustrated by the inequality 3 7 >3 2.
Properties of powers with integer exponents
Since positive integers are natural numbers, then all the properties of powers with positive integer exponents coincide exactly with the properties of powers with natural exponents listed and proven in the previous paragraph.
We defined a degree with an integer negative exponent, as well as a degree with a zero exponent, in such a way that all properties of degrees with natural exponents, expressed by equalities, remained valid. Therefore, all these properties are valid for both zero exponents and negative exponents, while, of course, the bases of the powers are different from zero.
So, for any real and non-zero numbers a and b, as well as any integers m and n, the following are true: properties of powers with integer exponents:
For a=0, the powers a m and a n make sense only when both m and n are positive integers, that is, natural numbers. Thus, the properties just written are also valid for the cases when a=0 and the numbers m and n are positive integers.
It is not difficult to prove each of these properties, for this it is enough to use the definitions of the degree with a natural and integer exponent, as well as the properties of actions with real numbers. As an example, let's prove that the power property holds for both positive integers and nonpositive integers. To do this, we need to show that if p is zero or a natural number and q is zero or a natural number, then the equalities (a p) q =a p q , (a − p) q =a (−p) q , (a p ) −q =a p·(−q) and (a −p) −q =a (−p)·(−q) . Let's do it.
For positive p and q, the equality (a p) q =a p·q was proven in the previous paragraph. If p=0, then we have (a 0) q =1 q =1 and a 0·q =a 0 =1, whence (a 0) q =a 0·q. Similarly, if q=0, then (a p) 0 =1 and a p·0 =a 0 =1, whence (a p) 0 =a p·0. If both p=0 and q=0, then (a 0) 0 =1 0 =1 and a 0·0 =a 0 =1, whence (a 0) 0 =a 0·0.
Now we prove that (a −p) q =a (−p)·q . By definition of a power with a negative integer exponent, then 
. By the property of quotients to powers we have 
. Since 1 p =1·1·…·1=1 and , then . The last expression is, by definition, a power of the form a −(p q) , which, by virtue of the multiplication rules, can be written as a (−p) q .
Likewise .
AND .
By the same principle, one can prove all other properties of a degree with an integer exponent, written in the form of equalities.
In the penultimate of the recorded properties, it is worth dwelling on the proof of the inequality a −n >b −n , which is true for any negative integer −n and any positive a and b for which the condition a . Let us write down and transform the difference between the left and right sides of this inequality: 
. Since by condition a n n , therefore, b n −a n >0 . The product a n ·b n is also positive as the product of positive numbers a n and b n . Then the resulting fraction is positive as a quotient of positive numbers b n − a n and a n b n . Therefore, whence a −n >b −n , which is what needed to be proved.
The last property of powers with integer exponents is proven in the same way as a similar property of powers with natural exponents.
Properties of powers with rational exponents
We defined a degree with a fractional exponent by extending the properties of a degree with an integer exponent to it. In other words, powers with fractional exponents have the same properties as powers with integer exponents. Namely:
- property of the product of powers with the same bases
for a>0, and if and, then for a≥0; - property of quotient powers with the same bases
for a>0 ; - property of a product to a fractional power
for a>0 and b>0, and if and, then for a≥0 and (or) b≥0; - property of a quotient to a fractional power
for a>0 and b>0, and if , then for a≥0 and b>0; - property of degree to degree
for a>0, and if and, then for a≥0; - property of comparing powers with equal rational exponents: for any positive numbers a and b, a 0 the inequality a p p is true, and for p p >b p ;
- the property of comparing powers with rational exponents and equal bases: for rational numbers p and q, p>q for 0p q, and for a>0 – inequality a p >a q.
- a p ·a q =a p+q ;
- a p:a q =a p−q ;
- (a·b) p =a p ·b p ;
- (a:b) p =a p:b p ;
- (a p) q =a p·q ;
- for any positive numbers a and b, a 0 the inequality a p p is true, and for p p >b p ;
- for irrational numbers p and q, p>q for 0p q, and for a>0 – the inequality a p >a q.
The proof of the properties of powers with fractional exponents is based on the definition of a power with a fractional exponent, on the properties of the arithmetic root of the nth degree and on the properties of a power with an integer exponent. Let us provide evidence.
By definition of a power with a fractional exponent and , then . The properties of the arithmetic root allow us to write the following equalities. Further, using the property of a degree with an integer exponent, we obtain , from which, by the definition of a degree with a fractional exponent, we have 
, and the indicator of the degree obtained can be transformed as follows: . This completes the proof.
The second property of powers with fractional exponents is proved in an absolutely similar way:
The remaining equalities are proved using similar principles:
Let's move on to proving the next property. Let us prove that for any positive a and b, a 0 the inequality a p p is true, and for p p >b p . Let's write the rational number p as m/n, where m is an integer and n is a natural number. The conditions p 0 in this case will be equivalent to the conditions m 0, respectively. For m>0 and am m . From this inequality, by the property of roots, we have, and since a and b are positive numbers, then, based on the definition of a degree with a fractional exponent, the resulting inequality can be rewritten as, that is, a p p .
Similarly, for m m >b m , whence, that is, a p >b p .
It remains to prove the last of the listed properties. Let us prove that for rational numbers p and q, p>q for 0p q, and for a>0 – the inequality a p >a q. We can always reduce rational numbers p and q to a common denominator, even if we get ordinary fractions and , where m 1 and m 2 are integers, and n is a natural number. In this case, the condition p>q will correspond to the condition m 1 >m 2, which follows from the rule for comparing ordinary fractions with the same denominators. Then, by the property of comparing degrees with the same bases and natural exponents, for 0m 1 m 2, and for a>1, the inequality a m 1 >a m 2. These inequalities in the properties of the roots can be rewritten accordingly as 
And 
. And the definition of a degree with a rational exponent allows us to move on to inequalities and, accordingly. From here we draw the final conclusion: for p>q and 0p q , and for a>0 – the inequality a p >a q .
Properties of powers with irrational exponents
From the way a degree with an irrational exponent is defined, we can conclude that it has all the properties of degrees with rational exponents. So for any a>0, b>0 and irrational numbers p and q the following are true properties of powers with irrational exponents:
From this we can conclude that powers with any real exponents p and q for a>0 have the same properties.
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In the last video lesson, we learned that the degree of a certain base is an expression that represents the product of the base by itself, taken in an amount equal to the exponent. Let us now study some of the most important properties and operations of powers.
For example, let's multiply two different powers with the same base:
Let's present this work in its entirety:
(2) 3 * (2) 2 = (2)*(2)*(2)*(2)*(2) = 32
Having calculated the value of this expression, we get the number 32. On the other hand, as can be seen from the same example, 32 can be represented as the product of the same base (two), taken 5 times. And indeed, if you count it, then:
Thus, we can confidently conclude that:
(2) 3 * (2) 2 = (2) 5
This rule works successfully for any indicators and any reasons. This property of power multiplication follows from the rule that the meaning of expressions is preserved during transformations in a product. For any base a, the product of two expressions (a)x and (a)y is equal to a(x + y). In other words, when any expressions with the same base are produced, the resulting monomial has a total degree formed by adding the degrees of the first and second expressions.
The presented rule also works great when multiplying several expressions. The main condition is that everyone has the same bases. For example:
(2) 1 * (2) 3 * (2) 4 = (2) 8
It is impossible to add degrees, and indeed to carry out any power-based joint actions with two elements of an expression if their bases are different.
As our video shows, due to the similarity of the processes of multiplication and division, the rules for adding powers in a product are perfectly transferred to the division procedure. Consider this example:
Let's transform the expression term by term into its full form and reduce the same elements in the dividend and divisor:
(2)*(2)*(2)*(2)*(2)*(2) / (2)*(2)*(2)*(2) = (2)(2) = (2) 2 = 4
The end result of this example is not so interesting, because already in the process of solving it it is clear that the value of the expression is equal to the square of two. And it is two that is obtained by subtracting the degree of the second expression from the degree of the first.
To determine the degree of the quotient, it is necessary to subtract the degree of the divisor from the degree of the dividend. The rule works with the same base for all its values and for all natural powers. In the form of abstraction we have:
(a) x / (a) y = (a) x - y
From the rule of dividing identical bases with degrees, the definition for the zero degree follows. Obviously, the following expression looks like:
(a) x / (a) x = (a) (x - x) = (a) 0
On the other hand, if we do the division in a more visual way, we get:
(a) 2 / (a) 2 = (a) (a) / (a) (a) = 1
When reducing all visible elements of a fraction, the expression 1/1 is always obtained, that is, one. Therefore, it is generally accepted that any base raised to the zero power is equal to one:
Regardless of the value of a.
However, it would be absurd if 0 (which still gives 0 for any multiplication) is somehow equal to one, so an expression of the form (0) 0 (zero to the zero power) simply does not make sense, and to formula (a) 0 = 1 add a condition: “if a is not equal to 0.”
Let's solve the exercise. Let's find the value of the expression:
(34) 7 * (34) 4 / (34) 11
Since the base is the same everywhere and equal to 34, the final value will have the same base with a degree (according to the above rules):
In other words:
(34) 7 * (34) 4 / (34) 11 = (34) 0 = 1
Answer: the expression is equal to one.
Expressions, expression conversion
Power expressions (expressions with powers) and their transformation
In this article we will talk about converting expressions with powers. First, we will focus on transformations that are performed with expressions of any kind, including power expressions, such as opening parentheses and bringing similar terms. And then we will analyze the transformations inherent specifically in expressions with degrees: working with the base and exponent, using the properties of degrees, etc.
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What are power expressions?
The term “power expressions” practically does not appear in school mathematics textbooks, but it appears quite often in collections of problems, especially those intended for preparation for the Unified State Exam and the Unified State Exam, for example. After analyzing the tasks in which it is necessary to perform any actions with power expressions, it becomes clear that power expressions are understood as expressions containing powers in their entries. Therefore, you can accept the following definition for yourself:
Definition.
Power expressions are expressions containing degrees.
Let's give examples of power expressions. Moreover, we will represent them according to how the development of views on from a degree with a natural indicator to a degree with a real indicator takes place.
As you know, first you get acquainted with the degree of a number with a natural exponent, at this stage the first simplest power expressions of the type 3 2 , 7 5 +1 , (2+1) 5 , (−0,1) 4 , 3 a 2 −a+a 2 , x 3−1 , (a 2) 3 etc.
A little later, the power of a number with an integer exponent is studied, which leads to the appearance of power expressions with negative integer powers, like the following: 3 −2, 
, a −2 +2 b −3 +c 2 .
In high school they return to degrees. There a degree with a rational exponent is introduced, which entails the appearance of the corresponding power expressions: 
, , 
and so on. Finally, degrees with irrational exponents and expressions containing them are considered: , .
The matter is not limited to the listed power expressions: further the variable penetrates into the exponent, and, for example, the following expressions arise: 2 x 2 +1 or 
. And after getting acquainted with , expressions with powers and logarithms begin to appear, for example, x 2·lgx −5·x lgx.
So, we have dealt with the question of what power expressions represent. Next we will learn to convert them.
Main types of transformations of power expressions
With power expressions, you can perform any of the basic identity transformations of expressions. For example, you can open parentheses, replace numerical expressions with their values, add similar terms, etc. Naturally, in this case, it is necessary to follow the accepted procedure for performing actions. Let's give examples.
Example.
Calculate the value of the power expression 2 3 ·(4 2 −12) .
Solution.
According to the order of execution of actions, first perform the actions in brackets. There, firstly, we replace the power 4 2 with its value 16 (if necessary, see), and secondly, we calculate the difference 16−12=4. We have 2 3 ·(4 2 −12)=2 3 ·(16−12)=2 3 ·4.
In the resulting expression, we replace the power 2 3 with its value 8, after which we calculate the product 8·4=32. This is the desired value.
So, 2 3 ·(4 2 −12)=2 3 ·(16−12)=2 3 ·4=8·4=32.
Answer:
2 3 ·(4 2 −12)=32.
Example.
Simplify expressions with powers 3 a 4 b −7 −1+2 a 4 b −7.
Solution.
Obviously, this expression contains similar terms 3·a 4 ·b −7 and 2·a 4 ·b −7 , and we can present them: .
Answer:
3 a 4 b −7 −1+2 a 4 b −7 =5 a 4 b −7 −1.
Example.
Express an expression with powers as a product.
Solution.
You can cope with the task by representing the number 9 as a power of 3 2 and then using the formula for abbreviated multiplication - difference of squares: 
Answer:
There are also a number of identical transformations inherent specifically in power expressions. We will analyze them further.
Working with base and exponent
There are degrees whose base and/or exponent are not just numbers or variables, but some expressions. As an example, we give the entries (2+0.3·7) 5−3.7 and (a·(a+1)−a 2) 2·(x+1) .
When working with such expressions, you can replace both the expression in the base of the degree and the expression in the exponent with an identically equal expression in the ODZ of its variables. In other words, according to the rules known to us, we can separately transform the base of the degree and separately the exponent. It is clear that as a result of this transformation, an expression will be obtained that is identically equal to the original one.
Such transformations allow us to simplify expressions with powers or achieve other goals we need. For example, in the power expression mentioned above (2+0.3 7) 5−3.7, you can perform operations with the numbers in the base and exponent, which will allow you to move to the power 4.1 1.3. And after opening the brackets and bringing similar terms to the base of the degree (a·(a+1)−a 2) 2·(x+1), we obtain a power expression of a simpler form a 2·(x+1) .
Using Degree Properties
One of the main tools for transforming expressions with powers is equalities that reflect . Let us recall the main ones. For any positive numbers a and b and arbitrary real numbers r and s, the following properties of powers are true:
- a r ·a s =a r+s ;
- a r:a s =a r−s ;
- (a·b) r =a r ·b r ;
- (a:b) r =a r:b r ;
- (a r) s =a r·s .
Note that for natural, integer, and positive exponents, the restrictions on the numbers a and b may not be so strict. For example, for natural numbers m and n the equality a m ·a n =a m+n is true not only for positive a, but also for negative a, and for a=0.
At school, the main focus when transforming power expressions is on the ability to choose the appropriate property and apply it correctly. In this case, the bases of degrees are usually positive, which allows the properties of degrees to be used without restrictions. The same applies to the transformation of expressions containing variables in the bases of powers - the range of permissible values of variables is usually such that the bases take only positive values on it, which allows you to freely use the properties of powers. In general, you need to constantly ask yourself whether it is possible to use any property of degrees in this case, because inaccurate use of properties can lead to a narrowing of the educational value and other troubles. These points are discussed in detail and with examples in the article transformation of expressions using properties of powers. Here we will limit ourselves to considering a few simple examples.
Example.
Express the expression a 2.5 ·(a 2) −3:a −5.5 as a power with base a.
Solution.
First, we transform the second factor (a 2) −3 using the property of raising a power to a power: (a 2) −3 =a 2·(−3) =a −6. The original power expression will take the form a 2.5 ·a −6:a −5.5. Obviously, it remains to use the properties of multiplication and division of powers with the same base, we have
a 2.5 ·a −6:a −5.5 =
a 2.5−6:a −5.5 =a −3.5:a −5.5 =
a −3.5−(−5.5) =a 2 .
Answer:
a 2.5 ·(a 2) −3:a −5.5 =a 2.
Properties of powers when transforming power expressions are used both from left to right and from right to left.
Example.
Find the value of the power expression.
Solution.
The equality (a·b) r =a r ·b r, applied from right to left, allows us to move from the original expression to a product of the form and further. And when multiplying powers with the same bases, the exponents add up: 
.
It was possible to transform the original expression in another way: 
Answer:
.
Example.
Given the power expression a 1.5 −a 0.5 −6, introduce a new variable t=a 0.5.
Solution.
The degree a 1.5 can be represented as a 0.5 3 and then, based on the property of the degree to the degree (a r) s =a r s, applied from right to left, transform it to the form (a 0.5) 3. Thus, a 1.5 −a 0.5 −6=(a 0.5) 3 −a 0.5 −6. Now it’s easy to introduce a new variable t=a 0.5, we get t 3 −t−6.
Answer:
t 3 −t−6 .
Converting fractions containing powers
Power expressions can contain or represent fractions with powers. Any of the basic transformations of fractions that are inherent in fractions of any kind are fully applicable to such fractions. That is, fractions that contain powers can be reduced, reduced to a new denominator, worked separately with their numerator and separately with the denominator, etc. To illustrate these words, consider solutions to several examples.
Example.
Simplify power expression 
.
Solution.
This power expression is a fraction. Let's work with its numerator and denominator. In the numerator we open the brackets and simplify the resulting expression using the properties of powers, and in the denominator we present similar terms: 
And let’s also change the sign of the denominator by placing a minus in front of the fraction: 
.
Answer:
.
Reducing fractions containing powers to a new denominator is carried out similarly to reducing rational fractions to a new denominator. In this case, an additional factor is also found and the numerator and denominator of the fraction are multiplied by it. When performing this action, it is worth remembering that reduction to a new denominator can lead to a narrowing of the VA. To prevent this from happening, it is necessary that the additional factor does not go to zero for any values of the variables from the ODZ variables for the original expression.
Example.
Reduce the fractions to a new denominator: a) to denominator a, b) 
to the denominator.
Solution.
a) In this case, it is quite easy to figure out which additional multiplier helps to achieve the desired result. This is a multiplier of a 0.3, since a 0.7 ·a 0.3 =a 0.7+0.3 =a. Note that in the range of permissible values of the variable a (this is the set of all positive real numbers), the power of a 0.3 does not vanish, therefore, we have the right to multiply the numerator and denominator of a given fraction by this additional factor: 
b) Taking a closer look at the denominator, you will find that 
and multiplying this expression by will give the sum of cubes and , that is, . And this is the new denominator to which we need to reduce the original fraction.
This is how we found an additional factor. In the range of acceptable values of the variables x and y, the expression does not vanish, therefore, we can multiply the numerator and denominator of the fraction by it: 
Answer:
A) 
, b) 
.
There is also nothing new in reducing fractions containing powers: the numerator and denominator are represented as a number of factors, and the same factors of the numerator and denominator are reduced.
Example.
Reduce the fraction: a) 
, b) .
Solution.
a) Firstly, the numerator and denominator can be reduced by the numbers 30 and 45, which is equal to 15. It is also obviously possible to perform a reduction by x 0.5 +1 and by 
. Here's what we have: 
b) In this case, identical factors in the numerator and denominator are not immediately visible. To obtain them, you will have to perform preliminary transformations. In this case, they consist in factoring the denominator using the difference of squares formula: 
Answer:
A) 
b) 
.
Converting fractions to a new denominator and reducing fractions are mainly used to do things with fractions. Actions are performed according to known rules. When adding (subtracting) fractions, they are reduced to a common denominator, after which the numerators are added (subtracted), but the denominator remains the same. The result is a fraction whose numerator is the product of the numerators, and the denominator is the product of the denominators. Division by a fraction is multiplication by its inverse.
Example.
Follow the steps 
.
Solution.
First, we subtract the fractions in parentheses. To do this, we bring them to a common denominator, which is 
, after which we subtract the numerators: 
Now we multiply the fractions: 
Obviously, it is possible to reduce by a power of x 1/2, after which we have 
.
You can also simplify the power expression in the denominator by using the difference of squares formula: 
.
Answer:
Example.
Simplify the Power Expression 
.
Solution.
Obviously, this fraction can be reduced by (x 2.7 +1) 2, this gives the fraction 
. It is clear that something else needs to be done with the powers of X. To do this, we transform the resulting fraction into a product. This gives us the opportunity to take advantage of the property of dividing powers with the same bases: 
. And at the end of the process we move from the last product to the fraction.
Answer:
.
And let us also add that it is possible, and in many cases desirable, to transfer factors with negative exponents from the numerator to the denominator or from the denominator to the numerator, changing the sign of the exponent. Such transformations often simplify further actions. For example, a power expression can be replaced by .
Converting expressions with roots and powers
Often, in expressions in which some transformations are required, roots with fractional exponents are also present along with powers. To transform such an expression to the desired form, in most cases it is enough to go only to roots or only to powers. But since it is more convenient to work with degrees, they usually move from roots to degrees. However, it is advisable to carry out such a transition when the ODZ of variables for the original expression allows you to replace the roots with powers without the need to refer to the module or split the ODZ into several intervals (we discussed this in detail in the article transition from roots to powers and back After getting acquainted with the degree with a rational exponent a degree with an irrational exponent is introduced, which allows us to talk about a degree with an arbitrary real exponent. At this stage, the school begins to study exponential function, which is analytically given by a power, the base of which is a number, and the exponent is a variable. So we are faced with power expressions containing numbers in the base of the power, and in the exponent - expressions with variables, and naturally the need arises to perform transformations of such expressions.
It should be said that the transformation of expressions of the indicated type usually has to be performed when solving exponential equations And exponential inequalities, and these conversions are quite simple. In the overwhelming majority of cases, they are based on the properties of the degree and are aimed, for the most part, at introducing a new variable in the future. The equation will allow us to demonstrate them 5 2 x+1 −3 5 x 7 x −14 7 2 x−1 =0.
Firstly, powers, in the exponents of which is the sum of a certain variable (or expression with variables) and a number, are replaced by products. This applies to the first and last terms of the expression on the left side:
5 2 x 5 1 −3 5 x 7 x −14 7 2 x 7 −1 =0,
5 5 2 x −3 5 x 7 x −2 7 2 x =0.
Next, both sides of the equality are divided by the expression 7 2 x, which on the ODZ of the variable x for the original equation takes only positive values (this is a standard technique for solving equations of this type, we are not talking about it now, so focus on subsequent transformations of expressions with powers ): 
Now we can cancel fractions with powers, which gives 
.
Finally, the ratio of powers with the same exponents is replaced by powers of relations, resulting in the equation 
, which is equivalent 
. The transformations made allow us to introduce a new variable, which reduces the solution of the original exponential equation to the solution of a quadratic equation
