Addition of degrees with the same exponent. Rules for multiplying degrees with different bases

We remind you that this lesson understands power properties with natural indicators and zero. Degrees with rational indicators and their properties will be discussed in the lessons for grade 8.

A natural exponent has several important properties that make it easier to calculate in exponent examples.

Property number 1
Product of degrees

Remember!

When multiplying degrees with the same bases, the base remains unchanged, and the exponents are added.

a m · a n \u003d a m + n, where "a" is any number, and "m", "n" are any natural numbers.

This property of degrees also affects the product of three or more degrees.

Simplify the expression.
b b 2 b 3 b 4 b 5 \u003d b 1 + 2 + 3 + 4 + 5 \u003d b 15
Present as a degree.
6 15 36 \u003d 6 15 6 2 \u003d 6 15 6 2 \u003d 6 17
Present as a degree.
(0.8) 3 (0.8) 12 \u003d (0.8) 3 + 12 \u003d (0.8) 15

Important!

Please note that in the specified property it was only about the multiplication of powers with on the same grounds ... It does not apply to their addition.

You cannot replace the amount (3 3 + 3 2) with 3 5. This is understandable if
count (3 3 + 3 2) \u003d (27 + 9) \u003d 36, and 3 5 \u003d 243

Property number 2
Private degrees

Remember!

When dividing degrees with the same bases, the base remains unchanged, and the exponent of the divisor is subtracted from the exponent of the dividend.

\u003d 11 3 - 2 4 2 - 1 \u003d 11 4 \u003d 44

Example. Solve the equation. We use the property of private degrees.
3 8: t \u003d 3 4

T \u003d 3 8 - 4

Answer: t \u003d 3 4 \u003d 81

Using properties # 1 and # 2, you can easily simplify expressions and perform calculations.

Example. Simplify the expression.
4 5m + 6 4 m + 2: 4 4m + 3 \u003d 4 5m + 6 + m + 2: 4 4m + 3 \u003d 4 6m + 8 - 4m - 3 \u003d 4 2m + 5
Example. Find the value of an expression using the properties of the degree.
= = = 2 9 + 2
2 5
= 2 11
2 5
= 2 11 − 5 = 2 6 = 64
Important!
Note that property 2 was only about dividing degrees with the same bases.

You cannot replace the difference (4 3 −4 2) with 4 1. This is understandable if you count (4 3 −4 2) = (64 − 16) = 48 , and 4 1 \u003d 4

Be careful!

Property number 3
Exponentiation

Remember!
When raising a degree to a power, the base of the degree remains unchanged, and the exponents are multiplied.
(a n) m \u003d a n · m, where "a" is any number, and "m", "n" are any natural numbers.

Properties 4
Degree of work

Remember!
When raising to the power of a product, each of the factors is raised to a power. The results are then multiplied.
(a · b) n \u003d a n · b n, where “a”, “b” are any rational numbers; "N" is any natural number.
- Example 1.
  (6 a 2 b 3 s) 2 \u003d 6 2 a 2 2 b 3 2 s 1 2 \u003d 36 a 4 b 6 s 2
- Example 2.
  (−x 2 y) 6 \u003d ((−1) 6 x 2 6 y 1 6) \u003d x 12 y 6
Important!
Note that property # 4, like other degree properties, is applied in reverse order.
(a n b n) \u003d (a b) n
That is, in order to multiply the powers with the same indicators, you can multiply the bases, and the exponent can be left unchanged.
- Example. Calculate.
  2 4 5 4 \u003d (2 5) 4 \u003d 10 4 \u003d 10,000
- Example. Calculate.
  0.5 16 2 16 \u003d (0.5 2) 16 \u003d 1
In more complex examples, there may be cases when multiplication and division must be performed over degrees with different bases and different exponents. In this case, we advise you to proceed as follows.

For example, 4 5 3 2 \u003d 4 3 4 2 3 2 \u003d 4 3 (4 3) 2 \u003d 64 12 2 \u003d 64 144 \u003d 9216

An example of raising to a decimal power.
4 21 (−0.25) 20 \u003d 4 4 20 (−0.25) 20 \u003d 4 (4 (−0.25)) 20 \u003d 4 (−1) 20 \u003d 4 1 \u003d 4
Properties 5
Degree of quotient (fraction)

Remember!
To raise the quotient to the power, you can raise the dividend and divisor separately to this power, and divide the first result by the second.
(a: b) n \u003d a n: b n, where “a”, “b” are any rational numbers, b ≠ 0, n is any natural number.
- Example. Present the expression in the form of private degrees.
  (5: 3) 12 = 5 12: 3 12
We remind you that the 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.

Aside from the eighth degree, what do we see here? We recall the 7th grade program. So, remember? This is the formula for abbreviated multiplication, namely the difference of squares! We get:

Let's take a close look at the denominator. It is very similar to one of the numerator factors, but what is wrong? Wrong order of terms. If they were to be reversed, the rule could be applied.

But how to do that? It turns out to be very easy: an even degree of the denominator helps us here.

The terms are magically reversed. This "phenomenon" is applicable to any expression to an even degree: we can freely change the signs in brackets.

But it's important to remember: all signs change at the same time!

Let's go back to the example:

And again the formula:

Whole we call the natural numbers opposite to them (that is, taken with the sign "") and the number.

positive integer, but it is no different from natural, then everything looks exactly like in the previous section.

Now let's look at some new cases. Let's start with an indicator equal to.

Any number in the zero degree is equal to one:

As always, let's ask ourselves: why is this so?

Consider some degree with a base. Take, for example, and multiply by:

So, we multiplied the number by, and we got the same as it was -. And what number should you multiply so that nothing changes? That's right, on. Means.

We can do the same with an arbitrary number:

Let's repeat the rule:

Any number in the zero degree is equal to one.

But there are exceptions to many rules. And here it is also there - this is a number (as a base).

On the one hand, it should be equal to any degree - no matter how much you multiply by yourself, you still get zero, this is clear. But on the other hand, like any number in the zero degree, it must equal. So which of this is true? Mathematicians decided not to get involved and refused to raise zero to zero. That is, now we cannot not only divide by zero, but also raise it to zero power.

Let's go further. In addition to natural numbers and numbers, negative numbers belong to integers. To understand what a negative power is, let's do the same as last time: multiply some normal number by the same negative power:

From here it is already easy to express what you are looking for:

Now we extend the resulting rule to an arbitrary degree:

So, let's formulate a rule:

A number in negative power is inverse to the same number in a positive power. But at the same time the base cannot be null: (because you cannot divide by).

Let's summarize:

I. Expression not specified in case. If, then.

II. Any number to the zero degree is equal to one:.

III. A number that is not zero is in negative power inverse to the same number in a positive power:.

Tasks for an independent solution:

Well, as usual, examples for an independent solution:

Analysis of tasks for independent solution:

I know, I know, the numbers are terrible, but on the exam you have to be ready for anything! Solve these examples or analyze their solution if you could not solve and you will learn how to easily cope with them on the exam!

Let's continue to expand the circle of numbers "suitable" as an exponent.

Now consider rational numbers. What numbers are called rational?

Answer: all that can be represented as a fraction, where and are integers, moreover.

To understand what is Fractional degree, consider the fraction:

Let's raise both sides of the equation to the power:

Now let's remember the rule about "Degree to degree":

What number must be raised to a power to get?

This formulation is the definition of the th root.

Let me remind you: the root of the th power of a number () is a number that, when raised to a power, is equal to.

That is, the th root is the inverse of the exponentiation:.

Turns out that. Obviously, this particular case can be extended:.

Now we add the numerator: what is it? The answer is easily obtained using the degree-to-degree rule:

But can the base be any number? After all, the root can not be extracted from all numbers.

None!

Remember the rule: any number raised to an even power is a positive number. That is, you cannot extract roots of an even degree from negative numbers!

This means that such numbers cannot be raised to a fractional power with an even denominator, that is, the expression has no meaning.

What about expression?

But here comes the problem.

The number can be represented as other, cancellable fractions, for example, or.

And it turns out that it exists, but does not exist, but these are just two different records of the same number.

Or another example: once, then you can write. But if we write down the indicator in a different way, and again we get a nuisance: (that is, we got a completely different result!).

To avoid such paradoxes, consider only positive radix with fractional exponent.

So if:

- natural number;
- an integer;

Examples:

Rational exponents are very useful for converting rooted expressions, for example:

5 examples to train

Analysis of 5 examples for training

1. Do not forget about the usual properties of degrees:

2.. Here we remember that we forgot to learn the table of degrees:

after all - it is or. The solution is found automatically:.

And now the hardest part. Now we will analyze irrational degree.

All the rules and properties of degrees here are exactly the same as for a degree with a rational exponent, with the exception of

Indeed, by definition, irrational numbers are numbers that cannot be represented as a fraction, where and are whole numbers (that is, irrational numbers are all real numbers except rational ones).

When studying degrees with a natural, whole and rational indicator, each time we made up a kind of "image", "analogy", or description in more familiar terms.

For example, a natural exponent is a number multiplied by itself several times;

...zero power number - it is, as it were, a number multiplied by itself once, that is, it has not yet begun to be multiplied, which means that the number itself has not even appeared - therefore, the result is only a kind of "blank number", namely the number;

...negative integer degree - it was as if some kind of "reverse process" took place, that is, the number was not multiplied by itself, but divided.

By the way, in science, a degree with a complex indicator is often used, that is, the indicator is not even a real number.

But at school we do not think about such difficulties, you will have the opportunity to comprehend these new concepts at the institute.

WHERE WE ARE SURE YOU WILL GO! (if you learn how to solve such examples :))

For example:

Decide for yourself:

Analysis of solutions:

1. Let's start with the already usual rule for raising a power to a power:

Now look at the indicator. Does he remind you of anything? We recall the formula for reduced multiplication, the difference of squares:

In this case,

Turns out that:

Answer: .

2. We bring fractions in exponents to the same form: either both decimals, or both ordinary. Let's get, for example:

Answer: 16

3. Nothing special, we apply the usual properties of degrees:

ADVANCED LEVEL

Determination of the degree

A degree is an expression of the form:, where:

— base of degree;
- exponent.

Degree with natural exponent (n \u003d 1, 2, 3, ...)

Raising a number to a natural power n means multiplying the number by itself times:

Integer degree (0, ± 1, ± 2, ...)

If the exponent is whole positive number:

Erection to zero:

The expression is indefinite, because, on the one hand, to any degree - this, and on the other - any number to the th degree - this.

If the exponent is whole negative number:

(because you cannot divide by).

Once again about zeros: expression is undefined in case. If, then.

Examples:

Rational grade

- natural number;
- an integer;

Examples:

Power properties

To make it easier to solve problems, let's try to understand: where did these properties come from? Let's prove them.

Let's see: what is and?

A-priory:

So, on the right side of this expression, we get the following product:

But by definition it is the power of a number with an exponent, that is:

Q.E.D.

Example : Simplify the expression.

Decision : .

Example : Simplify the expression.

Decision : It is important to note that in our rule necessarilymust have the same bases. Therefore, we combine the degrees with the base, but remains a separate factor:

One more important note: this rule is - only for the product of degrees!

By no means should I write that.

Just as with the previous property, let us turn to the definition of the degree:

Let's rearrange this piece like this:

It turns out that the expression is multiplied by itself once, that is, according to the definition, this is the th power of the number:

In essence, this can be called "bracketing the indicator". But you should never do this in total:!

Let's remember the abbreviated multiplication formulas: how many times did we want to write? But this is not true, after all.

A degree with a negative base.

Up to this point, we have only discussed how it should be index degree. But what should be the foundation? In degrees with natural indicator the basis can be any number .

Indeed, we can multiply any numbers by each other, be they positive, negative, or even. Let's think about which signs ("" or "") will have powers of positive and negative numbers?

For example, will the number be positive or negative? AND? ?

With the first, everything is clear: no matter how many positive numbers we multiply by each other, the result will be positive.

But negative is a little more interesting. After all, we remember a simple rule from the 6th grade: “minus by minus gives plus”. That is, or. But if we multiply by (), we get -.

And so on to infinity: with each subsequent multiplication, the sign will change. You can formulate such simple rules:

even degree, - number positive.
Negative number raised to odd degree, - number negative.
A positive number to any degree is a positive number.
Zero to any power is zero.

Decide on your own which sign the following expressions will have:

1.	2.	3.
4.	5.	6.

Did you manage? Here are the answers:

1) ; 2) ; 3) ; 4) ; 5) ; 6) .

In the first four examples, I hope everything is clear? We just look at the base and exponent and apply the appropriate rule.

In example 5), everything is also not as scary as it seems: it does not matter what the base is equal to - the degree is even, which means that the result will always be positive. Well, except when the base is zero. The foundation is not equal, is it? Obviously not, since (because).

Example 6) is no longer so simple. Here you need to find out which is less: or? If you remember that, it becomes clear that, which means that the base is less than zero. That is, we apply rule 2: the result will be negative.

And again we use the definition of degree:

Everything is as usual - we write down the definition of degrees and, divide them into each other, divide them into pairs and get:

Before examining the last rule, let's solve a few examples.

Calculate the values \u200b\u200bof the expressions:

Solutions :

Aside from the eighth degree, what do we see here? We recall the 7th grade program. So, remember? This is the formula for abbreviated multiplication, namely the difference of squares!

We get:

Let's take a close look at the denominator. It is very similar to one of the numerator factors, but what is wrong? Wrong order of terms. If they were swapped, Rule 3 could be applied. But how to do it? It turns out to be very easy: an even degree of the denominator helps us here.

If you multiply it by, nothing changes, right? But now it turns out the following:

The terms are magically reversed. This "phenomenon" is applicable to any expression to an even degree: we can freely change the signs in brackets. But it's important to remember: all signs change simultaneously!It cannot be replaced with by changing only one disadvantage that we do not want!

Let's go back to the example:

And again the formula:

So now the last rule:

How are we going to prove it? Of course, as usual: let's expand the concept of degree and simplify:

Now let's open the brackets. How many letters will there be? times by multipliers - what does it look like? This is nothing more than a definition of an operation multiplication: there were only multipliers. That is, it is, by definition, the degree of a number with an exponent:

Example:

Irrational grade

In addition to the information about the degrees for the intermediate level, we will analyze the degree with an irrational exponent. All the rules and properties of degrees here are exactly the same as for a degree with a rational exponent, with the exception - after all, by definition, irrational numbers are numbers that cannot be represented as a fraction, where and are integers (that is, irrational numbers are all real numbers, except rational).

When studying degrees with a natural, whole and rational indicator, each time we made up a kind of "image", "analogy", or description in more familiar terms. For example, a natural exponent is a number multiplied by itself several times; a number to the zero degree is, as it were, a number multiplied by itself once, that is, it has not yet begun to be multiplied, which means that the number itself has not even appeared - therefore, the result is only a kind of "blank number", namely the number; a degree with an integer negative exponent is as if a certain "reverse process" took place, that is, the number was not multiplied by itself, but divided.

It is extremely difficult to imagine a degree with an irrational exponent (just as it is difficult to imagine 4-dimensional space). Rather, it is a purely mathematical object that mathematicians created to extend the concept of a degree to the entire space of numbers.

By the way, in science, a degree with a complex indicator is often used, that is, the indicator is not even a real number. But at school we do not think about such difficulties, you will have the opportunity to comprehend these new concepts at the institute.

So what do we do when we see an irrational exponent? We are trying with all our might to get rid of it! :)

For example:

Decide for yourself:

Answers:

We recall the formula for the difference of squares. Answer:.
We bring fractions to the same form: either both decimals, or both ordinary. We get, for example:.
Nothing special, we apply the usual power properties:

SUMMARY OF THE SECTION AND BASIC FORMULAS

Degree called an expression of the form:, where:

Integer Degree

degree, the exponent of which is a natural number (i.e. whole and positive).

Rational grade

degree, the exponent of which is negative and fractional numbers.

Irrational grade

degree, the exponent of which is an infinite decimal fraction or root.

Power properties

Features of degrees.

Negative number raised to even degree, - number positive.
Negative number raised to odd degree, - number negative.
A positive number to any degree is a positive number.
Zero is equal to any degree.
Any number to the zero degree is equal.

NOW YOUR WORD ...

How do you like the article? Write down in the comments if you liked it or not.

Tell us about your experience with degree properties.

Perhaps you have questions. Or suggestions.

Write in the comments.

And good luck with your exams!

Science and Mathematics Articles

Properties of degrees with the same base

There are three properties of degrees with the same bases and natural values. it

Composition sum

Private two degrees with the same bases is equal to the expression, where the base is the same, and the exponent is difference indicators of the original factors.

Raising the power of a number to a power is equal to an expression in which the base is the same number and the exponent is composition two degrees.

Be careful! Rules regarding addition and subtraction degrees with the same bases does not exist.

Let's write these properties-rules in the form of formulas:

a m × a n \u003d a m + n

a m ÷ a n \u003d a m – n

(a m) n \u003d a mn

Now we will consider them with specific examples and try to prove them.

5 2 × 5 3 \u003d 5 5 - here we applied the rule; Now let's imagine how we would solve this example if we didn't know the rules:

5 2 × 5 3 \u003d 5 × 5 × 5 × 5 × 5 \u003d 5 5 - five squared is five times five, and cubed is the product of three fives. The result is the product of five fives, but this is something other than five to the fifth power: 5 5.

3 9 ÷ 3 5 \u003d 3 9–5 \u003d 3 4. Let's write the division as a fraction:

It can be shortened:

As a result, we get:

Thus, we proved that when dividing two degrees with the same bases, their indicators must be subtracted.

However, when dividing, you cannot have the divisor equal to zero (since you cannot divide by zero). In addition, since we consider degrees only with natural exponents, we cannot get a number less than 1 as a result of subtracting exponents. Therefore, restrictions are imposed on the formula a m ÷ a n \u003d a m – n: a ≠ 0 and m\u003e n.

Let's move on to the third property:
(2 2) 4 \u003d 2 2 × 4 \u003d 2 8

Let's write in expanded form:
(2 2) 4 \u003d (2 × 2) 4 \u003d (2 × 2) × (2 × 2) × (2 × 2) × (2 × 2) \u003d 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 \u003d 2 8

You can come to this conclusion and reasoning logically. You need to multiply two squared four times. But in each square there are two twos, which means there will be eight twos in total.

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Degree properties

We remind you that this lesson understands power properties with natural indicators and zero. Degrees with rational indicators and their properties will be discussed in the lessons for grade 8.

A natural exponent has several important properties that make it easier to calculate in exponent examples.

Property number 1
Product of degrees

When multiplying degrees with the same bases, the base remains unchanged, and the exponents are added.

a m · a n \u003d a m + n, where "a" is any number, and "m", "n" are any natural numbers.

This property of degrees also affects the product of three or more degrees.

Simplify the expression.
b b 2 b 3 b 4 b 5 \u003d b 1 + 2 + 3 + 4 + 5 \u003d b 15

Present as a degree.
6 15 36 \u003d 6 15 6 2 \u003d 6 15 6 2 \u003d 6 17

Present as a degree.
(0.8) 3 (0.8) 12 \u003d (0.8) 3 + 12 \u003d (0.8) 15

Please note that in the specified property, it was only about the multiplication of powers with the same bases ... It does not apply to their addition.

You cannot replace the amount (3 3 + 3 2) with 3 5. This is understandable if
count (3 3 + 3 2) \u003d (27 + 9) \u003d 36, and 3 5 \u003d 243

Property number 2
Private degrees

When dividing degrees with the same bases, the base remains unchanged, and the exponent of the divisor is subtracted from the exponent of the dividend.

Write the quotient as a degree
(2b) 5: (2b) 3 \u003d (2b) 5 - 3 \u003d (2b) 2

Calculate.

11 3 - 2 4 2 - 1 \u003d 11 4 \u003d 44
Example. Solve the equation. We use the property of private degrees.
3 8: t \u003d 3 4

Answer: t \u003d 3 4 \u003d 81

Using properties # 1 and # 2, you can easily simplify expressions and perform calculations.

Example. Find the value of an expression using the properties of the degree.

2 11 − 5 = 2 6 = 64

Note that property 2 was only about dividing degrees with the same bases.

You cannot replace the difference (4 3 −4 2) with 4 1. This is understandable if you count (4 3 −4 2) \u003d (64 - 16) \u003d 48, and 4 1 \u003d 4

Property number 3
Exponentiation

When raising a degree to a power, the base of the degree remains unchanged, and the exponents are multiplied.

(a n) m \u003d a n · m, where "a" is any number, and "m", "n" are any natural numbers.

Note that property # 4, like other degree properties, is applied in reverse order.

(a n b n) \u003d (a b) n

That is, in order to multiply the powers with the same indicators, you can multiply the bases, and the exponent can be left unchanged.

Example. Calculate.
2 4 5 4 \u003d (2 5) 4 \u003d 10 4 \u003d 10,000

Example. Calculate.
0.5 16 2 16 \u003d (0.5 2) 16 \u003d 1

In more complex examples, there may be cases when multiplication and division must be performed over degrees with different bases and different exponents. In this case, we advise you to proceed as follows.

For example, 4 5 3 2 \u003d 4 3 4 2 3 2 \u003d 4 3 (4 3) 2 \u003d 64 12 2 \u003d 64 144 \u003d 9216

An example of raising to a decimal power.

4 21 (−0.25) 20 \u003d 4 4 20 (−0.25) 20 \u003d 4 (4 (−0.25)) 20 \u003d 4 (−1) 20 \u003d 4 1 \u003d 4

Properties 5
Degree of quotient (fraction)

To raise the quotient to the power, you can raise the dividend and divisor separately to this power, and divide the first result by the second.

(a: b) n \u003d a n: b n, where “a”, “b” are any rational numbers, b ≠ 0, n is any natural number.

Example. Present the expression in the form of private degrees.
(5: 3) 12 = 5 12: 3 12

We remind you that the 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.

Multiplying and dividing numbers with powers

If you need to raise a specific number to a power, you can use the table of powers of natural numbers from 2 to 25 in algebra. And now we will dwell on properties of degrees.

Exponential numbers they open up great possibilities, they allow us to transform multiplication into addition, and adding is much easier than multiplying.

For example, we need to multiply 16 by 64. The product of the multiplication of these two numbers is 1024. But 16 is 4x4, and 64 is 4x4x4. That is, 16 by 64 \u003d 4x4x4x4x4, which is also 1024.

The number 16 can also be represented as 2x2x2x2, and 64 as 2x2x2x2x2x2, and if we multiply, we again get 1024.

Now we use the rule for raising a number to a power. 16 \u003d 4 2, or 2 4, 64 \u003d 4 3, or 2 6, at the same time 1024 \u003d 6 4 \u003d 4 5, or 2 10.

Therefore, our problem can be written in another way: 4 2 x4 3 \u003d 4 5 or 2 4 x2 6 \u003d 2 10, and each time we get 1024.

We can solve some similar examples and see that multiplying numbers with powers reduces to addition of exponents, or exponential, of course, provided that the bases of the factors are equal.

Thus, without multiplying, we can immediately say that 2 4 x2 2 x2 14 \u003d 2 20.

This rule is also true when dividing numbers with powers, but in this case, e the exponent of the divisor is subtracted from the exponent of the dividend... Thus, 2 5: 2 3 \u003d 2 2, which in ordinary numbers is 32: 8 \u003d 4, that is, 2 2. Let's summarize:

a m х a n \u003d a m + n, a m: a n \u003d a m-n, where m and n are integers.

At first glance, it may seem what is multiplication and division of numbers with powers not very convenient, because first you need to represent the number in exponential form. It is not difficult to represent the numbers 8 and 16 in this form, that is, 2 3 and 2 4, but how to do this with the numbers 7 and 17? Or what to do when the number can be represented in exponential form, but the bases of the exponential expressions of numbers are very different. For example, 8 x 9 is 2 3 x 3 2, in which case we cannot sum the exponents. Neither 2 5 nor 3 5 is the answer, nor does the answer lie between these two numbers.

Then is it worth bothering with this method at all? Definitely worth it. It offers tremendous benefits, especially for complex and time consuming computations.

Until now, we have assumed that the exponent is the number of identical factors. In this case, the minimum value of the exponent is 2. However, if we perform the operation of dividing numbers, or subtracting exponentials, we can also get a number less than 2, which means that the old definition can no longer suit us. Read more in the next article.

Addition, subtraction, multiplication, and division of powers

Add and subtract powers

Obviously, numbers with powers can be added, like other quantities , by adding them one by one 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 equal degrees of identical variables can be added or subtracted.

So, the sum of 2a 2 and 3a 2 is 5a 2.

It is also obvious that if you take two squares a, or three squares a, or five squares a.

But degrees different variables and varying degrees identical variables, must be added by their addition 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 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 degrees is carried out in the same way as addition, except that the signs of the subtracted must be changed accordingly.

Or:
2a 4 - (-6a 4) \u003d 8a 4
3h 2 b 6 - 4h 2 b 6 \u003d -h 2 b 6
5 (a - h) 6 - 2 (a - h) 6 \u003d 3 (a - h) 6

Multiplication of degrees

Numbers with powers can be multiplied, like other quantities, by writing them one after the other, with or without a multiplication sign between them.

So, the result of multiplying a 3 by b 2 is a 3 b 2 or aaabb.

Or:
x -3 ⋅ a m \u003d a m x -3
3a 6 y 2 ⋅ (-2x) \u003d -6a 6 xy 2
a 2 b 3 y 2 ⋅ a 3 b 2 y \u003d a 2 b 3 y 2 a 3 b 2 y

The result in the last example can be ordered by adding the same 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 sum degrees of terms.

So, a 2 .a 3 \u003d aa.aaa \u003d aaaaa \u003d a 5.

Here 5 is the power of the multiplication result, equal to 2 + 3, the sum of the powers of the terms.

So, a n .a m \u003d a m + n.

For a n, a is taken as a factor as many times as the power of n is equal;

And a m, is taken as a factor as many times as the power of m is;

Therefore, degrees with the same stems can be multiplied by adding the exponents.

So, a 2 .a 6 \u003d a 2 + 6 \u003d a 8. And x 3 .x 2 .x \u003d x 3 + 2 + 1 \u003d x 6.

Or:
4a n ⋅ 2a n \u003d 8a 2n
b 2 y 3 ⋅ b 4 y \u003d b 6 y 4
(b + h - y) n ⋅ (b + h - y) \u003d (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 \u003d a -5. This can be written as (1 / aa). (1 / aaa) \u003d 1 / aaaaa.

2.y -n .y -m \u003d y -n-m.

3.a -n .a m \u003d a m-n.

If a + b is multiplied by a - b, the result is 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 the sum and the difference of two numbers raised to square, the result will be equal to the sum or difference of these numbers in fourth degree.

So, (a - y). (A + y) \u003d a 2 - y 2.
(a 2 - y 2) ⋅ (a 2 + y 2) \u003d a 4 - y 4.
(a 4 - y 4) ⋅ (a 4 + y 4) \u003d a 8 - y 8.

Division of degrees

Power numbers can be divided, like other numbers, by subtracting from the divisor, or by placing them in fractional form.

So a 3 b 2 divided by b 2 equals a 3.

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 difference exponents of divisible numbers.

When dividing degrees with the same base, their indicators are subtracted..

So, y 3: y 2 \u003d y 3-2 \u003d y 1. That is, $ \\ frac \u003d y $.

And a n + 1: a \u003d a n + 1-1 \u003d a n. That is, $ \\ frac \u003d a ^ n $.

Or:
y 2m: y m \u003d y m
8a n + m: 4a m \u003d 2a n
12 (b + y) n: 3 (b + y) 3 \u003d 4 (b + y) n-3

The rule is also true for numbers with negative values \u200b\u200bof degrees.
The result of dividing a -5 by a -3 is a -2.
Also, $ \\ frac: \\ frac \u003d \\ frac. \\ Frac \u003d \\ frac \u003d \\ frac $.

h 2: h -1 \u003d h 2 + 1 \u003d h 3 or $ h ^ 2: \\ frac \u003d h ^ 2. \\ frac \u003d 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 exponents in $ \\ frac $ Answer: $ \\ frac $.

2. Decrease the exponents in $ \\ frac $. Answer: $ \\ frac $ or 2x.

3. Decrease the exponents a 2 / a 3 and a -3 / a -4 and bring them to the common denominator.
a 2 .a -4 is a -2 first numerator.
a 3 .a -3 is a 0 \u003d 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. Decrease the exponents 2a 4 / 5a 3 and 2 / a 4 and bring them to the 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.

The degree and its properties. Middle level.

Do you want to test your strength and find out the result of how ready you are for the Unified State Exam or the OGE?

Degree called an expression of the form:, where:

Integer Degree

degree, the exponent of which is a natural number (i.e. whole and positive).

Rational grade

degree, the exponent of which is negative and fractional numbers.

Irrational grade

degree, the exponent of which is an infinite decimal fraction or root.

Power properties

Features of degrees.

even degree, - number positive.

Negative number raised to odd degree, - number negative.

A positive number to any degree is a positive number.

Zero is equal to any degree.

Any number to the zero degree is equal.

What is the degree of a number?

Exponentiation is the same mathematical operation as addition, subtraction, multiplication, or division.

Now I will explain everything in human language using very simple examples. Pay attention. The examples are elementary, but they explain important things.

Let's start with addition.

There is nothing to explain. You already know everything: there are eight of us. Each has two bottles of cola. How much cola is there? That's right - 16 bottles.

Now multiplication.

The same cola example can be written differently:. Mathematicians are cunning and lazy people. They first notice some patterns, and then come up with a way to quickly "count" them. In our case, they noticed that each of the eight people had the same number of cola bottles and came up with a technique called multiplication. Agree, it is considered easier and faster than.

So, to count faster, easier and without errors, you just need to remember multiplication table... You can, of course, do everything slower, harder and with mistakes! But…

Here is the multiplication table. Repeat.

And another, more beautiful:

What other clever counting tricks have lazy mathematicians come up with? Right - raising a number to a power.

Raising a number to a power.

If you need to multiply a number by itself five times, then mathematicians say that you need to raise this number to the fifth power. For example, . Mathematicians remember that two to the fifth degree is. And they solve such problems in their head - faster, easier and without mistakes.

All you need to do is remember what is highlighted in the table of powers of numbers... Believe me, this will make your life much easier.

By the way, why is the second degree called square numbers, and the third - cube? What does it mean? That's a very good question. Now you will have both squares and cubes.

An example from life # 1.

Let's start with a square or the second power of a number.

Imagine a square meter by meter pool. The pool is in your country house. It's hot and I really want to swim. But ... a pool without a bottom! You need to cover the bottom of the pool with tiles. How many tiles do you need? In order to determine this, you need to know the area of \u200b\u200bthe pool bottom.

You can just count, poking your finger, that the bottom of the pool is made up of meter by meter cubes. If you have a tile meter by meter, you will need pieces. It's easy ... But where have you seen such tiles? The tile is more likely to be cm by cm. And then you will be tormented by the "finger count". Then you have to multiply. So, on one side of the bottom of the pool, we will fit tiles (pieces) and on the other, too, tiles. Multiplying by, you get tiles ().

Have you noticed that we multiplied the same number by itself to determine the area of \u200b\u200bthe pool bottom? What does it mean? Once the same number is multiplied, we can use the "exponentiation" technique. (Of course, when you have only two numbers, you still multiply them or raise them to a power. But if you have a lot of them, then raising to a power is much easier and there are also fewer errors in calculations. For the exam, this is very important).
So, thirty in the second degree will be (). Or you can say that thirty squared will be. In other words, the second power of a number can always be represented as a square. And vice versa, if you see a square, it is ALWAYS the second power of a number. A square is an image of the second power of a number.

Real life example # 2.

Here's a task for you, count how many squares are on the chessboard using the square of the number. On one side of the cells and on the other too. To count their number, you need to multiply eight by eight, or ... if you notice that the chessboard is a square with a side, then you can square eight. You will get cells. () So?

Life example # 3.

Now the cube or the third power of the number. The same pool. But now you need to find out how much water will have to be poured into this pool. You need to calculate the volume. (Volumes and liquids, by the way, are measured in cubic meters. Surprisingly, right?) Draw a pool: the bottom is a meter in size and a meter deep, and try to calculate how many cubic meters by meter will go into your pool.

Point your finger and count! One, two, three, four ... twenty two, twenty three ... How much did it turn out? Not lost? Is it difficult to count with your finger? So that! Take an example from mathematicians. They are lazy, so they noticed that in order to calculate the volume of the pool, you need to multiply its length, width and height by each other. In our case, the volume of the pool will be equal to cubes ... Easier, right?

Now imagine how lazy and cunning mathematicians are if they simplified this too. They reduced everything to one action. They noticed that the length, width and height are equal and that the same number is multiplied by itself ... What does that mean? This means that you can use the degree. So, what you once counted with your finger, they do in one action: three in a cube is equal. It is written like this:.

Only remains remember the table of degrees... If you, of course, are as lazy and cunning as mathematicians. If you like to work hard and make mistakes, you can continue to count with your finger.

Well, to finally convince you that the degrees were invented by idlers and cunning people to solve their life problems, and not to create problems for you, here are a couple more examples from life.

Life example # 4.

You have a million rubles. At the beginning of each year, you make another million from every million. That is, your every million at the beginning of each year doubles. How much money will you have in years? If you are now sitting and “counting with your finger,” then you are a very hardworking person and .. stupid. But most likely you will give an answer in a couple of seconds, because you are smart! So, in the first year - two times two ... in the second year - what happened was two more, in the third year ... Stop! You noticed that the number is multiplied by itself once. So two to the fifth power is a million! Now imagine that you have a competition and those millions will be received by the one who calculates faster ... Is it worth remembering the degrees of numbers, what do you think?

Real life example # 5.

You have a million. At the beginning of each year, you earn two more on every million. Great, isn't it? Every million triples. How much money will you have in years? Let's count. The first year - multiply by, then the result by another ... It's already boring, because you already understood everything: three times is multiplied by itself. So the fourth power is equal to a million. You just need to remember that three to the fourth power is or.

Now you know that by raising a number to a power, you will greatly facilitate your life. Let's take a closer look at what you can do with degrees and what you need to know about them.

Terms and concepts.

So, first, let's define the concepts. What do you think, what is exponent? It is very simple - this is the number that is "at the top" of the power of the number. Not scientific, but understandable and easy to remember ...

Well, at the same time that such degree basis? Even simpler is the number that is at the bottom, at the base.

Here's a drawing to be sure.

Well, in general terms, in order to generalize and remember better ... A degree with a base "" and an indicator "" is read as "in degree" and is written as follows:

"The degree of a number with a natural exponent"

You probably already guessed: because the exponent is a natural number. Yes, but what is natural number? Elementary! Natural numbers are those numbers that are used in counting when listing items: one, two, three ... When we count items, we do not say: "minus five", "minus six", "minus seven". We also do not say: "one third", or "zero point, five tenths." These are not natural numbers. What numbers do you think?

Numbers like “minus five”, “minus six”, “minus seven” refer to whole numbers. In general, whole numbers include all natural numbers, numbers opposite to natural numbers (that is, taken with a minus sign), and a number. Zero is easy to understand - this is when there is nothing. And what do negative ("minus") numbers mean? But they were invented primarily to indicate debts: if you have rubles on your phone, it means that you owe the operator rubles.

Any fractions are rational numbers. How do you think they came about? Very simple. Several thousand years ago, our ancestors discovered that they lacked natural numbers to measure length, weight, area, etc. And they came up with rational numbers... Interesting, right?

There are also irrational numbers. What are these numbers? In short, an infinite decimal fraction. For example, if you divide the circumference of a circle by its diameter, you get an irrational number.

Natural numbers are numbers used for counting, that is, etc.

Integers - all natural numbers, natural numbers with a minus and the number 0.

Fractional numbers are considered rational.

Irrational numbers are an infinite decimal fraction

Degree with a natural indicator

Let's define the concept of a degree, the exponent of which is a natural number (that is, an integer and positive).

Any number in the first power is equal to itself:
To square a number is to multiply it by itself:
To cube a number is to multiply it by itself three times:

Definition. Raising a number to a natural power means multiplying the number by itself times:

If you need to raise a specific number to a power, you can use. And now we will dwell on properties of degrees.

Exponential numbers they open up great possibilities, they allow us to transform multiplication into addition, and adding is much easier than multiplying.

For example, we need to multiply 16 by 64. The product of the multiplication of these two numbers is 1024. But 16 is 4x4, and 64 is 4x4x4. That is, 16 by 64 \u003d 4x4x4x4x4, which is also 1024.

The number 16 can also be represented as 2x2x2x2, and 64 as 2x2x2x2x2x2, and if we multiply, we again get 1024.

Now let's use the rule. 16 \u003d 4 2, or 2 4, 64 \u003d 4 3, or 2 6, at the same time 1024 \u003d 6 4 \u003d 4 5, or 2 10.