How to raise a product and a quotient to a power. Exponentiation of product and quotient

If we do not pay attention to the eighth degree, what do we see here? Let's take a look at the 7th grade program. So, remember? This is the abbreviated multiplication formula, namely the difference of squares! We get:

We carefully look at the denominator. It looks a lot like one of the numerator factors, but what's wrong? Wrong order of terms. If they were swapped, the rule could apply.

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

The terms have magically changed places. This "phenomenon" applies 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 name the natural numbers, their opposites (that is, taken with the sign "") and the number.

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

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

Any number to the zero power is equal to one:

As always, we ask ourselves: why is this so?

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

So, we multiplied the number by, and got the same as it was -. What number must be multiplied by 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 to the zero power 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 must be equal to any degree - no matter how much you multiply zero by itself, you still get zero, this is clear. But on the other hand, like any number to the zero degree, it must be equal. So what is the truth of this? Mathematicians decided not to get involved and refused to raise zero to the zero power. That is, now we can not only divide by zero, but also raise it to the zero power.

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

From here it is already easy to express the desired:

Now we extend the resulting rule to an arbitrary degree:

So, let's formulate the rule:

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

Let's summarize:

I. Expression is not defined in case. If, then.

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

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

Tasks for independent solution:

Well, as usual, examples for an independent solution:

Analysis of tasks for independent solution:

I know, I know, the numbers are scary, but at the exam you have to be ready for anything! Solve these examples or analyze their solution if you couldn't solve it and you will learn how to easily deal with them in 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" Let's consider a fraction:

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

Now remember the rule "degree to degree":

What number must be raised to a power to get?

This formulation is the definition of the root of the th degree.

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

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

It turns out that. Obviously, this special case can be extended: .

Now add the numerator: what is it? The answer is easy to get with the power-to-power 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, it is impossible to extract roots of an even degree from negative numbers!

And this means that such numbers cannot be raised to a fractional power with an even denominator, that is, the expression does not make sense.

What about expression?

But here a problem arises.

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

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

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

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

So if:

- natural number;
is an integer;

Examples:

Powers with a rational exponent are very useful for transforming expressions with roots, for example:

5 practice examples

Analysis of 5 examples for training

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

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

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

Well, now - the most difficult. Now we will analyze degree with an irrational exponent.

All the rules and properties of degrees here are exactly the same as for degrees 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 integers (that is, irrational numbers are all real numbers except rational ones).

When studying degrees with a natural, integer and rational indicator, each time we made up a certain “image”, “analogy”, or description in more familiar terms.

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

...zero power- this 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 yet - therefore the result is only a certain “number blank”, namely the number;

...negative integer exponent- it’s as if a certain “reverse process” has taken place, that is, the number was not multiplied by itself, but divided.

By the way, science often uses a degree with a complex exponent, that is, an exponent is not even a real number.

But at school, we don’t 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 degree to a degree:

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

In this case,

It turns out that:

Answer: .

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

Answer: 16

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

ADVANCED LEVEL

Definition of degree

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

— base of degree;
- exponent.

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

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

Power with integer exponent (0, ±1, ±2,...)

If the exponent is positive integer number:

erection to zero power:

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

If the exponent is integer negative number:

(because it is impossible to divide).

One more time about nulls: the expression is not defined in the case. If, then.

Examples:

Degree with rational exponent

- natural number;
is an integer;

Examples:

Degree 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?

By definition:

So, on the right side of this expression, the following product is obtained:

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

Q.E.D.

Example : Simplify the expression.

Solution : .

Example : Simplify the expression.

Solution : It is important to note that in our rule necessarily must have the same basis. Therefore, we combine the degrees with the base, but remain a separate factor:

Another important note: this rule - only for products of powers!

Under no circumstances should I write that.

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

Let's rearrange it 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 fact, this can be called "bracketing the indicator". But you can never do this in total:!

Let's recall the formulas for abbreviated multiplication: how many times did we want to write? But that's not true, really.

Power with a negative base.

Up to this point, we have discussed only what should be index degree. But what should be the basis? In degrees from natural indicator the basis may be any number .

Indeed, we can multiply any number by each other, whether they are positive, negative, or even. Let's think about what signs (" " or "") will have degrees of positive and negative numbers?

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

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

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

And so on ad infinitum: with each subsequent multiplication, the sign will change. It is possible to formulate such simple rules:

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

Determine for yourself what 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 simply look at the base and exponent, and apply the appropriate rule.

In example 5), everything is also not as scary as it seems: it doesn’t 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 base is not the same, 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 analyzing the last rule, let's solve a few examples.

Calculate the values of expressions:

Solutions :

We get:

We carefully look at the denominator. It looks a lot like one of the numerator factors, but what's wrong? Wrong order of terms. If they were reversed, rule 3 could be applied. But how to do this? It turns out that it is very easy: the even degree of the denominator helps us here.

If you multiply it by, nothing changes, right? But now it looks like this:

The terms have magically changed places. This "phenomenon" applies 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! It cannot be replaced by by changing only one objectionable minus to us!

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:

Well, now let's open the brackets. How many letters will there be? times by multipliers - what does it look like? This is nothing but the definition of an operation multiplication: total there turned out to be multipliers. That is, it is, by definition, a power of a number with an exponent:

Example:

Degree with irrational exponent

In addition to information about the degrees for the average level, we will analyze the degree with an irrational indicator. 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 ones).

When studying degrees with a natural, integer and rational indicator, each time we made up a certain “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 yet - therefore, the result is only a certain “preparation of a number”, namely a number; a degree with an integer negative indicator - it is as if a certain “reverse process” has occurred, 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 a 4-dimensional space). Rather, it is a purely mathematical object that mathematicians have created to extend the concept of a degree to the entire space of numbers.

By the way, science often uses a degree with a complex exponent, that is, an exponent is not even a real number. But at school, we don’t think about such difficulties; you will have the opportunity to comprehend these new concepts at the institute.

So what do we do if we see an irrational exponent? We are trying our best to get rid of it! :)

For example:

Decide for yourself:

Answers:

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

SECTION SUMMARY AND BASIC FORMULA

Degree is called an expression of the form: , where:

Degree with integer exponent

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

Degree with rational exponent

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

Degree with irrational exponent

exponent whose exponent is an infinite decimal fraction or root.

Degree 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 power is a positive number.
Zero is equal to any power.
Any number to the zero power is equal.

NOW YOU HAVE A WORD...

How do you like the article? Let me know in the comments below if you liked it or not.

Tell us about your experience with the power properties.

Perhaps you have questions. Or suggestions.

Write in the comments.

And good luck with your exams!

Please note that this section deals with the concept degrees only with a natural indicator and zero.

The concept and properties of degrees with rational exponents (with negative and fractional) will be discussed in lessons for grade 8.

So, let's figure out what a degree of a number is. To write the product of a number by itself, the abbreviated notation is used several times.

Instead of multiplying six identical factors 4 4 4 4 4 4 they write 4 6 and say "four to the sixth power."

4 4 4 4 4 4 = 4 6

The expression 4 6 is called the power of a number, where:

4 — base of degree;
6 — exponent.

In general, the degree with the base "a" and the exponent "n" is written using the expression:

Remember!

The degree of the number "a" with a natural exponent " n", greater than 1, is the product " n" identical factors, each of which is equal to the number "a".

The record " a n"It reads like this:" and to the power n "or" n-th power of the number a".

The exceptions are the entries:

a 2 - it can be pronounced as “a squared”;
a 3 - it can be pronounced as "a in a cube."

a 2 - "and to the second degree";
a 3 - "a to the third degree."

Special cases arise if the exponent is equal to one or zero (n = 1; n = 0).

Remember!

The degree of the number "a" with the exponent n \u003d 1 is this number itself:
a 1 = a

Any number to the zero power is equal to one.
a 0 = 1

Zero to any natural power is equal to zero.
0 n = 0

One to any power equals 1.
1n=1

Expression 0 0 ( zero to zero power) is considered meaningless.

(−32) 0 = 1
0 253 = 0
1 4 = 1

When solving examples, you need to remember that raising to a power is called finding a numeric or literal value after raising it to a power.

Example. Raise to a power.

5 3 = 5 5 5 = 125
2.5 2 = 2.5 2.5 = 6.25
( · = = 81
256

Exponentiation of a negative number

The base of the power (the number that is raised to a power) can be any number — positive, negative, or zero.

Remember!

Raising a positive number to a power results in a positive number.

Raising zero to a natural power results in zero.

When raising a negative number to a power, the result can be either a positive number or a negative number. It depends on whether the exponent was an even or odd number.

Consider examples of raising negative numbers to a power.

It can be seen from the examples considered that if a negative number is raised to an odd power, then a negative number is obtained. Since the product of an odd number of negative factors is negative.

If a negative number is raised to an even power, then a positive number is obtained. Since the product of an even number of negative factors is positive.

Remember!

A negative number raised to an even power is a positive number.

A negative number raised to an odd power is a negative number.

The square of any number is a positive number or zero, that is:

a 2 ≥ 0 for any a .

2 (−3) 2 = 2 (−3) (−3) = 2 9 = 18
−5 (−2) 3 = −5 (−8) = 40

Note!

When solving exponentiation examples, mistakes are often made, forgetting that the entries (−5) 4 and −5 4 are different expressions. The results of raising to a power of these expressions will be different.

Calculate (−5) 4 means to find the value of the fourth power of a negative number.

(−5) 4 = (−5) (−5) (−5) (−5) = 625

While finding "-5 4" means that the example needs to be solved in 2 steps:

Raise the positive number 5 to the fourth power.
5 4 = 5 5 5 5 = 625
Put a minus sign in front of the result obtained (that is, perform a subtraction action).
−5 4 = −625

Example. Calculate: −6 2 − (−1) 4

−6 2 − (−1) 4 = −37

6 2 = 6 6 = 36
−6 2 = −36
(−1) 4 = (−1) (−1) (−1) (−1) = 1
−(−1) 4 = −1
−36 − 1 = −37

Procedure for Examples with Degrees

Computing a value is called the action of exponentiation. This is the third stage action.

Remember!

In expressions with degrees that do not contain brackets, first perform exponentiation, then multiplication and division, and at the end addition and subtraction.

If there are brackets in the expression, then first, in the order indicated above, the actions in the brackets are performed, and then the remaining actions in the same order from left to right.

Example. Calculate:

To facilitate the solution of examples, it is useful to know and use the degree table, which you can download for free on our website.

To check your results, you can use the calculator on our website "

We figured out what the degree of a number is in general. Now we need to understand how to correctly calculate it, i.e. raise numbers to powers. In this material, we will analyze the basic rules for calculating the degree in the case of an integer, natural, fractional, rational and irrational exponent. All definitions will be illustrated with examples.

The concept of exponentiation

Let's start with the formulation of basic definitions.

Definition 1

Exponentiation is the calculation of the value of the power of some number.

That is, the words "calculation of the value of the degree" and "exponentiation" mean the same thing. So, if the task is "Raise the number 0 , 5 to the fifth power", this should be understood as "calculate the value of the power (0 , 5) 5 .

Now we give the basic rules that must be followed in such calculations.

Recall what a power of a number with a natural exponent is. For a power with base a and exponent n, this will be the product of the nth number of factors, each of which is equal to a. This can be written like this:

To calculate the value of the degree, you need to perform the operation of multiplication, that is, multiply the bases of the degree the specified number of times. The very concept of a degree with a natural indicator is based on the ability to quickly multiply. Let's give examples.

Example 1

Condition: Raise - 2 to the power of 4 .

Solution

Using the definition above, we write: (− 2) 4 = (− 2) (− 2) (− 2) (− 2) . Next, we just need to follow these steps and get 16 .

Let's take a more complicated example.

Example 2

Calculate the value 3 2 7 2

Solution

This entry can be rewritten as 3 2 7 · 3 2 7 . Earlier we looked at how to correctly multiply the mixed numbers mentioned in the condition.

Perform these steps and get the answer: 3 2 7 3 2 7 = 23 7 23 7 = 529 49 = 10 39 49

If the task indicates the need to raise irrational numbers to a natural power, we will need to first round their bases to a digit that will allow us to get an answer of the desired accuracy. Let's take an example.

Example 3

Perform the squaring of the number π .

Solution

Let's round it up to hundredths first. Then π 2 ≈ (3, 14) 2 = 9, 8596. If π ≈ 3 . 14159, then we will get a more accurate result: π 2 ≈ (3, 14159) 2 = 9, 8695877281.

Note that the need to calculate the powers of irrational numbers in practice arises relatively rarely. We can then write the answer as the power itself (ln 6) 3 or convert if possible: 5 7 = 125 5 .

Separately, it should be indicated what the first power of a number is. Here you can just remember that any number raised to the first power will remain itself:

This is clear from the record. .

It does not depend on the basis of the degree.

Example 4

So, (− 9) 1 = − 9 , and 7 3 raised to the first power remains equal to 7 3 .

For convenience, we will analyze three cases separately: if the exponent is a positive integer, if it is zero, and if it is a negative integer.

In the first case, this is the same as raising to a natural power: after all, positive integers belong to the set of natural numbers. We have already described how to work with such degrees above.

Now let's see how to properly raise to the zero power. With a base that is non-zero, this calculation always produces an output of 1 . We have previously explained that the 0th power of a can be defined for any real number not equal to 0 , and a 0 = 1 .

Example 5

5 0 = 1 , (- 2 , 56) 0 = 1 2 3 0 = 1

0 0 - not defined.

We are left with only the case of a degree with a negative integer exponent. We have already discussed that such degrees can be written as a fraction 1 a z, where a is any number, and z is a negative integer. We see that the denominator of this fraction is nothing more than an ordinary degree with a positive integer, and we have already learned how to calculate it. Let's give examples of tasks.

Example 6

Raise 2 to the -3 power.

Solution

Using the definition above, we write: 2 - 3 = 1 2 3

We calculate the denominator of this fraction and get 8: 2 3 \u003d 2 2 2 \u003d 8.

Then the answer is: 2 - 3 = 1 2 3 = 1 8

Example 7

Raise 1, 43 to the -2 power.

Solution

Reformulate: 1 , 43 - 2 = 1 (1 , 43) 2

We calculate the square in the denominator: 1.43 1.43. Decimals can be multiplied in this way:

As a result, we got (1, 43) - 2 = 1 (1, 43) 2 = 1 2 , 0449 . It remains for us to write this result in the form of an ordinary fraction, for which it is necessary to multiply it by 10 thousand (see the material on the conversion of fractions).

Answer: (1, 43) - 2 = 10000 20449

A separate case is raising a number to the minus first power. The value of such a degree is equal to the number opposite to the original value of the base: a - 1 \u003d 1 a 1 \u003d 1 a.

Example 8

Example: 3 − 1 = 1 / 3

9 13 - 1 = 13 9 6 4 - 1 = 1 6 4 .

How to raise a number to a fractional power

To perform such an operation, we need to recall the basic definition of a degree with a fractional exponent: a m n \u003d a m n for any positive a, integer m and natural n.

Definition 2

Thus, the calculation of a fractional degree must be performed in two steps: raising to an integer power and finding the root of the nth degree.

We have the equality a m n = a m n , which, given the properties of the roots, is usually used to solve problems in the form a m n = a n m . This means that if we raise the number a to a fractional power m / n, then first we extract the root of the nth degree from a, then we raise the result to a power with an integer exponent m.

Let's illustrate with an example.

Example 9

Calculate 8 - 2 3 .

Solution

Method 1. According to the basic definition, we can represent this as: 8 - 2 3 \u003d 8 - 2 3

Now let's calculate the degree under the root and extract the third root from the result: 8 - 2 3 = 1 64 3 = 1 3 3 64 3 = 1 3 3 4 3 3 = 1 4

Method 2. Let's transform the basic equality: 8 - 2 3 \u003d 8 - 2 3 \u003d 8 3 - 2

After that, we extract the root 8 3 - 2 = 2 3 3 - 2 = 2 - 2 and square the result: 2 - 2 = 1 2 2 = 1 4

We see that the solutions are identical. You can use any way you like.

There are cases when the degree has an indicator expressed as a mixed number or decimal fraction. For ease of calculation, it is better to replace it with an ordinary fraction and count as indicated above.

Example 10

Raise 44.89 to the power of 2.5.

Solution

Let's convert the value of the indicator into an ordinary fraction: 44 , 89 2 , 5 = 44 , 89 5 2 .

And now we perform all the actions indicated above in order: 44 , 89 5 2 = 44 , 89 5 = 44 , 89 5 = 4489 100 5 = 4489 100 5 = 67 2 10 2 5 = 67 10 5 = = 1350125107 100000 = 13 501, 25107

Answer: 13501, 25107.

If there are large numbers in the numerator and denominator of a fractional exponent, then calculating such exponents with rational exponents is a rather difficult job. It usually requires computer technology.

Separately, we dwell on the degree with a zero base and a fractional exponent. An expression of the form 0 m n can be given the following meaning: if m n > 0, then 0 m n = 0 m n = 0 ; if m n< 0 нуль остается не определен. Таким образом, возведение нуля в дробную положительную степень приводит к нулю: 0 7 12 = 0 , 0 3 2 5 = 0 , 0 0 , 024 = 0 , а в целую отрицательную - значения не имеет: 0 - 4 3 .

How to raise a number to an irrational power

The need to calculate the value of the degree, in the indicator of which there is an irrational number, does not arise so often. In practice, the task is usually limited to calculating an approximate value (up to a certain number of decimal places). This is usually calculated on a computer due to the complexity of such calculations, so we will not dwell on this in detail, we will only indicate the main provisions.

If we need to calculate the value of the degree a with an irrational exponent a , then we take the decimal approximation of the exponent and count from it. The result will be an approximate answer. The more accurate the decimal approximation taken, the more accurate the answer. Let's show with an example:

Example 11

Calculate the approximate value of 2 to the power of 1.174367....

Solution

We restrict ourselves to the decimal approximation a n = 1 , 17 . Let's do the calculations using this number: 2 1 , 17 ≈ 2 , 250116 . If we take, for example, the approximation a n = 1 , 1743 , then the answer will be a little more precise: 2 1 , 174367 . . . ≈ 2 1 . 1743 ≈ 2 . 256833 .

If you notice a mistake in the text, please highlight it and press Ctrl+Enter

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

An exponent with a natural exponent has several important properties that allow you to simplify calculations in exponent examples.

Property #1
Product of powers

Remember!

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

a m a n \u003d a m + n, where " a"- any number, and" m", " n"- any natural numbers.

This property of powers also affects 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 as a degree.
6 15 36 = 6 15 6 2 = 6 15 6 2 = 6 17
Present as a degree.
(0.8) 3 (0.8) 12 = (0.8) 3 + 12 = (0.8) 15

Important!

Please note that in the indicated property it was only about multiplying powers with the same grounds . 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 #2
Private degrees

Remember!

When dividing powers with the same base, 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 partial degrees.
3 8: t = 3 4

T = 3 8 − 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 degree properties.
= = = 2 9 + 2
2 5
= 2 11
2 5
= 2 11 − 5 = 2 6 = 64
Important!
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 we consider (4 3 −4 2) = (64 − 16) = 48 , and 4 1 = 4

Be careful!

Property #3
Exponentiation

Remember!
When raising a power to a power, the base of the power 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
Product degree

Remember!
When raising a product to a power, 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" - any natural number.
- Example 1
  (6 a 2 b 3 c) 2 = 6 2 a 2 2 b 3 2 s 1 2 = 36 a 4 b 6 s 2
- Example 2
  (−x 2 y) 6 = ((−1) 6 x 2 6 y 1 6) = x 12 y 6
Important!
Please note that property No. 4, like other properties of degrees, is also applied in reverse order.
(a n b n)= (a b) n
That is, to multiply degrees with the same exponents, you can multiply the bases, and leave the exponent unchanged.
- Example. Calculate.
  2 4 5 4 = (2 5) 4 = 10 4 = 10,000
- Example. Calculate.
  0.5 16 2 16 = (0.5 2) 16 = 1
In more complex examples, there may be cases when multiplication and division must be performed on powers with different bases and different exponents. In this case, we advise you to do the following.

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

Example of exponentiation of a decimal fraction.
4 21 (−0.25) 20 = 4 4 20 (−0.25) 20 = 4 (4 (−0.25)) 20 = 4 (−1) 20 = 4 1 = four
Properties 5
Power of the quotient (fractions)

Remember!
To raise a quotient to a 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. Express the expression as partial powers.
  (5: 3) 12 = 5 12: 3 12
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.

primary goal

To acquaint students with the properties of degrees with natural indicators and teach them to perform actions with degrees.

Topic “Degree and its properties” includes three questions:

Determination of the degree with a natural indicator.
Multiplication and division of powers.
Exponentiation of product and degree.

test questions

Formulate the definition of a degree with a natural exponent greater than 1. Give an example.
Formulate a definition of the degree with an indicator of 1. Give an example.
What is the order of operations when evaluating the value of an expression containing powers?
Formulate the main property of the degree. Give an example.
Formulate a rule for multiplying powers with the same base. Give an example.
Formulate a rule for dividing powers with the same bases. Give an example.
Formulate the rule for exponentiation of a product. Give an example. Prove the identity (ab) n = a n b n .
Formulate a rule for raising a degree to a power. Give an example. Prove the identity (a m) n = a m n .

Definition of degree.

degree of number a with a natural indicator n, greater than 1, is called the product of n factors, each of which is equal to a. degree of number a with exponent 1 the number itself is called a.

Degree with base a and indicator n is written like this: a n. It reads " a to the extent n”; “ n-th power of a number a ”.

By definition of degree:

a 4 = a a a a

. . . . . . . . . . . .

Finding the value of the degree is called exponentiation .

1. Examples of exponentiation:

3 3 = 3 3 3 = 27

0 4 = 0 0 0 0 = 0

(-5) 3 = (-5) (-5) (-5) = -125

25 ; 0,09 ;

25 = 5 2 ; 0,09 = (0,3) 2 ; .

27 ; 0,001 ; 8 .

27 = 3 3 ; 0,001 = (0,1) 3 ; 8 = 2 3 .

4. Find expression values:

a) 3 10 3 = 3 10 10 10 = 3 1000 = 3000

b) -2 4 + (-3) 2 = 7
2 4 = 16
(-3) 2 = 9
-16 + 9 = 7

Option 1

a) 0.3 0.3 0.3

c) b b b b b b b

d) (-x) (-x) (-x) (-x)

e) (ab) (ab) (ab)

2. Square the numbers:

3. Cube the numbers:

4. Find expression values:

c) -1 4 + (-2) 3

d) -4 3 + (-3) 2

e) 100 - 5 2 4

Multiplication of powers.

For any number a and arbitrary numbers m and n, the following is true:

a m a n = a m + n .

Proof:

rule : When multiplying powers with the same base, the bases remain the same, and the exponents are added.

a m a n a k = a m + n a k = a (m + n) + k = a m + n + k

a) x 5 x 4 = x 5 + 4 = x 9

b) y y 6 = y 1 y 6 = y 1 + 6 = y 7

c) b 2 b 5 b 4 \u003d b 2 + 5 + 4 \u003d b 11

d) 3 4 9 = 3 4 3 2 = 3 6

e) 0.01 0.1 3 = 0.1 2 0.1 3 = 0.1 5

a) 2 3 2 = 2 4 = 16

b) 3 2 3 5 = 3 7 = 2187

Option 1

1. Present as a degree:

a) x 3 x 4 e) x 2 x 3 x 4

b) a 6 a 2 g) 3 3 9

c) y 4 y h) 7 4 49

d) a a 8 i) 16 2 7

e) 2 3 2 4 j) 0.3 3 0.09

2. Present as a degree and find the value in the table:

a) 2 2 2 3 c) 8 2 5

b) 3 4 3 2 d) 27 243

Division of degrees.

For any number a0 and arbitrary natural numbers m and n such that m>n, the following holds:

a m: a n = a m - n

Proof:

a m - n a n = a (m - n) + n = a m - n + n = a m

by definition of private:

a m: a n \u003d a m - n.

rule: When dividing powers with the same base, the base is left the same, and the exponent of the divisor is subtracted from the exponent of the dividend.

Definition: The degree of a non-zero number with a zero exponent is equal to one:

because a n: a n = 1 for a0 .

a) x 4: x 2 \u003d x 4 - 2 \u003d x 2

b) y 8: y 3 = y 8 - 3 = y 5

c) a 7: a \u003d a 7: a 1 \u003d a 7 - 1 \u003d a 6

d) s 5:s 0 = s 5:1 = s 5

a) 5 7:5 5 = 5 2 = 25

b) 10 20:10 17 = 10 3 = 1000

in)

G)

e)

Option 1

1. Express the quotient as a power:

2. Find the values of expressions:

Raising to the power of a product.

For any a and b and an arbitrary natural number n:

(ab) n = a n b n

Proof:

By definition of degree

(ab) n =

Grouping the factors a and factors b separately, we get:

The proved property of the degree of the product extends to the degree of the product of three or more factors.

For example:

(a b c) n = a n b n c n ;

(a b c d) n = a n b n c n d n .

rule: When raising a product to a power, each factor is raised to that power and the result is multiplied.

1. Raise to a power:

a) (a b) 4 = a 4 b 4

b) (2 x y) 3 \u003d 2 3 x 3 y 3 \u003d 8 x 3 y 3

c) (3 a) 4 = 3 4 a 4 = 81 a 4

d) (-5 y) 3 \u003d (-5) 3 y 3 \u003d -125 y 3

e) (-0.2 x y) 2 \u003d (-0.2) 2 x 2 y 2 \u003d 0.04 x 2 y 2

f) (-3 a b c) 4 = (-3) 4 a 4 b 4 c 4 = 81 a 4 b 4 c 4

2. Find the value of the expression:

a) (2 10) 4 = 2 4 10 4 = 16 1000 = 16000

b) (3 5 20) 2 = 3 2 100 2 = 9 10000= 90000

c) 2 4 5 4 = (2 5) 4 = 10 4 = 10000

d) 0.25 11 4 11 = (0.25 4) 11 = 1 11 = 1

e)

Option 1

1. Raise to a power:

b) (2 a c) 4

e) (-0.1 x y) 3

2. Find the value of the expression:

b) (5 7 20) 2

Exponentiation.

For any number a and arbitrary natural numbers m and n:

(a m) n = a m n

Proof:

By definition of degree

(a m) n =

Rule: When raising a power to a power, the base is left the same, and the exponents are multiplied.

1. Raise to a power:

(a 3) 2 = a 6 (x 5) 4 = x 20

(y 5) 2 = y 10 (b 3) 3 = b 9

2. Simplify expressions:

a) a 3 (a 2) 5 = a 3 a 10 = a 13

b) (b 3) 2 b 7 \u003d b 6 b 7 \u003d b 13

c) (x 3) 2 (x 2) 4 \u003d x 6 x 8 \u003d x 14

d) (y y 7) 3 = (y 8) 3 = y 24