The product of numbers with different powers. Degree and its properties. An exhaustive guide (2020). Basic properties of degrees with irrational exponents

In the previous article, we talked about what monomials are. In this material, we will analyze how to solve examples and problems in which they are used. Here we will consider such actions as subtraction, addition, multiplication, division of monomials and raising them to a power with a natural exponent. We will show how such operations are defined, we will indicate the basic rules for their implementation and what should be the result. All theoretical provisions, as usual, will be illustrated by examples of problems with descriptions of solutions.

It is most convenient to work with the standard notation of monomials, so we present all the expressions that will be used in the article in a standard form. If they are initially set differently, it is recommended to first bring them to a generally accepted form.

Rules for adding and subtracting monomials

The simplest operations that can be performed with monomials are subtraction and addition. In the general case, the result of these actions will be a polynomial (a monomial is possible in some special cases).

When we add or subtract monomials, we first write down the corresponding sum and difference in the generally accepted form, after which we simplify the resulting expression. If there are similar terms, they must be given, the brackets must be opened. Let's explain with an example.

Example 1

Condition: add the monomials − 3 · x and 2 , 72 · x 3 · y 5 · z .

Solution

Let's write down the sum of the original expressions. Add parentheses and put a plus sign between them. We will get the following:

(− 3 x) + (2 , 72 x 3 y 5 z)

When we expand the brackets, we get - 3 x + 2 , 72 x 3 y 5 z . This is a polynomial, written in standard form, which will be the result of adding these monomials.

Answer:(− 3 x) + (2 , 72 x 3 y 5 z) = − 3 x + 2 , 72 x 3 y 5 z .

If we have three, four or more terms given, we perform this action in the same way.

Example 2

Condition: perform the given operations with polynomials in the correct order

3 a 2 - (- 4 a c) + a 2 - 7 a 2 + 4 9 - 2 2 3 a c

Solution

Let's start by opening parentheses.

3 a 2 + 4 a c + a 2 - 7 a 2 + 4 9 - 2 2 3 a c

We see that the resulting expression can be simplified by reducing like terms:

3 a 2 + 4 a c + a 2 - 7 a 2 + 4 9 - 2 2 3 a c = = (3 a 2 + a 2 - 7 a 2) + 4 a c - 2 2 3 a c + 4 9 = = - 3 a 2 + 1 1 3 a c + 4 9

We have a polynomial, which will be the result of this action.

Answer: 3 a 2 - (- 4 a c) + a 2 - 7 a 2 + 4 9 - 2 2 3 a c = - 3 a 2 + 1 1 3 a c + 4 9

In principle, we can perform the addition and subtraction of two monomials, with some restrictions, so that we end up with a monomial. To do this, it is necessary to observe some conditions regarding the terms and subtracted monomials. We will describe how this is done in a separate article.

Rules for multiplying monomials

The multiplication action does not impose any restrictions on multipliers. The monomials to be multiplied must not meet any additional conditions in order for the result to be a monomial.

To perform multiplication of monomials, you need to perform the following steps:

1. Record the piece correctly.
2. Expand the brackets in the resulting expression.
3. Group, if possible, factors with the same variables and numerical factors separately.
4. Perform the necessary actions with numbers and apply the property of multiplying powers with the same bases to the remaining factors.

Let's see how this is done in practice.

Example 3

Condition: multiply the monomials 2 · x 4 · y · z and - 7 16 · t 2 · x 2 · z 11 .

Solution

Let's start with the composition of the work.

Opening the brackets in it and we get the following:

2 x 4 y z - 7 16 t 2 x 2 z 11

2 - 7 16 t 2 x 4 x 2 y z 3 z 11

All we have to do is multiply the numbers in the first brackets and apply the power property to the second. As a result, we get the following:

2 - 7 16 t 2 x 4 x 2 y z 3 z 11 = - 7 8 t 2 x 4 + 2 y z 3 + 11 = = - 7 8 t 2 x 6 y z 14

Answer: 2 x 4 y z - 7 16 t 2 x 2 z 11 = - 7 8 t 2 x 6 y z 14 .

If we have three or more polynomials in the condition, we multiply them using exactly the same algorithm. We will consider the issue of multiplication of monomials in more detail in a separate material.

Rules for raising a monomial to a power

We know that the product of a certain number of identical factors is called a degree with a natural exponent. Their number is indicated by the number in the indicator. According to this definition, raising a monomial to a power is equivalent to multiplying the indicated number of identical monomials. Let's see how it's done.

Example 4

Condition: raise the monomial − 2 · a · b 4 to the power of 3 .

Solution

We can replace exponentiation with multiplication of 3 monomials − 2 · a · b 4 . Let's write down and get the desired answer:

(− 2 a b 4) 3 = (− 2 a b 4) (− 2 a b 4) (− 2 a b 4) = = ((− 2) (− 2) (− 2)) (a a a) (b 4 b 4 b 4) = − 8 a 3 b 12

Answer:(− 2 a b 4) 3 = − 8 a 3 b 12 .

But what about when the degree has a large exponent? Recording a large number of multipliers is inconvenient. Then, to solve such a problem, we need to apply the properties of the degree, namely the property of the degree of the product and the property of the degree in the degree.

Let's solve the problem that we cited above in the indicated way.

Example 5

Condition: raise − 2 · a · b 4 to the third power.

Solution

Knowing the property of the degree in the degree, we can proceed to an expression of the following form:

(− 2 a b 4) 3 = (− 2) 3 a 3 (b 4) 3 .

After that, we raise to the power - 2 and apply the exponent property:

(− 2) 3 (a) 3 (b 4) 3 = − 8 a 3 b 4 3 = − 8 a 3 b 12 .

Answer:− 2 · a · b 4 = − 8 · a 3 · b 12 .

We also devoted a separate article to raising a monomial to a power.

Rules for dividing monomials

The last action with monomials that we will analyze in this material is the division of a monomial by a monomial. As a result, we should get a rational (algebraic) fraction (in some cases, it is possible to obtain a monomial). Let us clarify right away that division by zero monomial is not defined, since division by 0 is not defined.

To perform division, we need to write the indicated monomials in the form of a fraction and reduce it, if possible.

Example 6

Condition: divide the monomial − 9 x 4 y 3 z 7 by − 6 p 3 t 5 x 2 y 2 .

Solution

Let's start by writing the monomials in the form of a fraction.

9 x 4 y 3 z 7 - 6 p 3 t 5 x 2 y 2

This fraction can be reduced. After doing this, we get:

3 x 2 y z 7 2 p 3 t 5

Answer:- 9 x 4 y 3 z 7 - 6 p 3 t 5 x 2 y 2 = 3 x 2 y z 7 2 p 3 t 5 .

The conditions under which, as a result of dividing monomials, we get a monomial are given in a separate article.

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Power formulas used in the process of reducing and simplifying complex expressions, in solving equations and inequalities.

Number c is an n-th power of a number a when:

Operations with degrees.

1. Multiplying degrees with the same base, their indicators add up:

a ma n = a m + n .

2. In the division of degrees with the same base, their indicators are subtracted:

3. The degree of the product of 2 or more factors is equal to the product of the degrees of these factors:

(abc…) n = a n b n c n …

4. The degree of a fraction is equal to the ratio of the degrees of the dividend and the divisor:

(a/b) n = a n / b n .

5. Raising a power to a power, the exponents are multiplied:

(am) n = a m n .

Each formula above is correct in the directions from left to right and vice versa.

For example. (2 3 5/15)² = 2² 3² 5²/15² = 900/225 = 4.

Operations with roots.

1. The root of the product of several factors is equal to the product of the roots of these factors:

2. The root of the ratio is equal to the ratio of the dividend and the divisor of the roots:

3. When raising a root to a power, it is enough to raise the root number to this power:

4. If we increase the degree of the root in n once and at the same time raise to n th power is a root number, then the value of the root will not change:

5. If we decrease the degree of the root in n root at the same time n th degree from the radical number, then the value of the root will not change:

Degree with a negative exponent. The degree of a certain number with a non-positive (integer) exponent is defined as one divided by the degree of the same number with an exponent equal to the absolute value of the non-positive exponent:

Formula a m:a n = a m - n can be used not only for m> n, but also at m< n.

For example. a4:a 7 = a 4 - 7 = a -3.

To formula a m:a n = a m - n became fair at m=n, you need the presence of the zero degree.

Degree with zero exponent. The power of any non-zero number with a zero exponent is equal to one.

For example. 2 0 = 1,(-5) 0 = 1,(-3/5) 0 = 1.

A degree with a fractional exponent. To raise a real number but to a degree m/n, you need to extract the root n th degree of m th power of this number but.

The concept of a degree in mathematics is introduced as early as the 7th grade in an algebra lesson. And in the future, throughout the course of studying mathematics, this concept is actively used in its various forms. Degrees are a rather difficult topic, requiring memorization of values ​​and the ability to correctly and quickly count. For faster and better work with mathematics degrees, they came up with the properties of a degree. They help to cut down on big calculations, to convert a huge example into a single number to some degree. There are not so many properties, and all of them are easy to remember and apply in practice. Therefore, the article discusses the main properties of the degree, as well as where they are applied.

degree properties

We will consider 12 properties of a degree, including properties of powers with the same base, and give an example for each property. Each of these properties will help you solve problems with degrees faster, as well as save you from numerous computational errors.

1st property.

Many people very often forget about this property, make mistakes, representing a number to the zero degree as zero.

2nd property.

3rd property.

It must be remembered that this property can only be used when multiplying numbers, it does not work with the sum! And we must not forget that this and the following properties apply only to powers with the same base.

4th property.

If the number in the denominator is raised to a negative power, then when subtracting, the degree of the denominator is taken in brackets to correctly replace the sign in further calculations.

The property only works when dividing, not when subtracting!

5th property.

6th property.

This property can also be applied in reverse. A unit divided by a number to some degree is that number to a negative power.

7th property.

This property cannot be applied to sum and difference! When raising a sum or difference to a power, abbreviated multiplication formulas are used, not the properties of the power.

8th property.

9th property.

This property works for any fractional degree with a numerator equal to one, the formula will be the same, only the degree of the root will change depending on the denominator of the degree.

Also, this property is often used in reverse order. The root of any power of a number can be represented as that number to the power of one divided by the power of the root. This property is very useful in cases where the root of the number is not extracted.

10th property.

This property works not only with the square root and the second degree. If the degree of the root and the degree to which this root is raised are the same, then the answer will be a radical expression.

11th property.

You need to be able to see this property in time when solving it in order to save yourself from huge calculations.

12th property.

Each of these properties will meet you more than once in tasks, it can be given in its pure form, or it may require some transformations and the use of other formulas. Therefore, for the correct solution, it is not enough to know only the properties, you need to practice and connect the rest of mathematical knowledge.

Application of degrees and their properties

They are actively used in algebra and geometry. Degrees in mathematics have a separate, important place. With their help, exponential equations and inequalities are solved, as well as powers often complicate equations and examples related to other sections of mathematics. Exponents help to avoid large and long calculations, it is easier to reduce and calculate the exponents. But to work with large powers, or with powers of large numbers, you need to know not only the properties of the degree, but also competently work with the bases, be able to decompose them in order to make your task easier. For convenience, you should also know the meaning of numbers raised to a power. This will reduce your time in solving by eliminating the need for long calculations.

The concept of degree plays a special role in logarithms. Since the logarithm, in essence, is the power of a number.

Abbreviated multiplication formulas are another example of the use of powers. They cannot use the properties of degrees, they are decomposed according to special rules, but in each abbreviated multiplication formula there are invariably degrees.

Degrees are also actively used in physics and computer science. All translations into the SI system are made using degrees, and in the future, when solving problems, the properties of the degree are applied. In computer science, powers of two are actively used, for the convenience of counting and simplifying the perception of numbers. Further calculations on conversions of units of measurement or calculations of problems, just like in physics, occur using the properties of the degree.

Degrees are also very useful in astronomy, where you can rarely find the use of the properties of a degree, but the degrees themselves are actively used to shorten the recording of various quantities and distances.

Degrees are also used in everyday life, when calculating areas, volumes, distances.

With the help of degrees, very large and very small values ​​\u200b\u200bare written in any field of science.

exponential equations and inequalities

Degree properties occupy a special place precisely in exponential equations and inequalities. These tasks are very common, both in the school course and in exams. All of them are solved by applying the properties of the degree. The unknown is always in the degree itself, therefore, knowing all the properties, it will not be difficult to solve such an equation or inequality.

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 range 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: .

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 “preparation of a number”, namely a 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, in science, a degree with a complex exponent is often used, 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,

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 indicator 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. You can formulate these simple rules:

1. even degree, - number positive.
2. Negative number raised to odd degree, - number negative.
3. A positive number to any power is a positive number.
4. 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 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 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 the 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 :

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 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 a negative integer - it's 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, in science, a degree with a complex exponent is often used, 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:

1)

2)

3)

Answers:

1. Remember the difference of squares formula. Answer: .
2. We bring fractions to the same form: either both decimals, or both ordinary ones. We get, for example: .
3. 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!

Lesson content

What is a degree?

Degree called the product of several identical factors. For example:

2×2×2

The value of this expression is 8

2 x 2 x 2 = 8

The left side of this equation can be made shorter - first write down the repeating factor and indicate over it how many times it repeats. The repeating multiplier in this case is 2. It repeats three times. Therefore, over the deuce, we write the triple:

2 3 = 8

This expression reads like this: two to the third power equals eight or " the third power of 2 is 8.

The short form of writing the multiplication of the same factors is used more often. Therefore, we must remember that if another number is inscribed over some number, then this is the multiplication of several identical factors.

For example, if the expression 5 3 is given, then it should be borne in mind that this expression is equivalent to writing 5 × 5 × 5.

The number that repeats is called base of degree. In the expression 5 3 the base of the degree is the number 5 .

And the number that is inscribed above the number 5 is called exponent. In the expression 5 3, the exponent is the number 3. The exponent shows how many times the base of the degree is repeated. In our case, base 5 is repeated three times.

The operation of multiplying identical factors is called exponentiation.

For example, if you need to find the product of four identical factors, each of which is equal to 2, then they say that the number 2 raised to the fourth power:

We see that the number 2 to the fourth power is the number 16.

Note that in this lesson we are looking at degrees with a natural indicator. This is a kind of degree, the exponent of which is a natural number. Recall that natural numbers are integers that are greater than zero. For example, 1, 2, 3 and so on.

In general, the definition of a degree with a natural indicator is as follows:

Degree of a with a natural indicator n is an expression of the form a n, which is equal to the product n multipliers, each of which is equal to a

Examples:

Be careful when raising a number to a power. Often, through inattention, a person multiplies the base of the degree by the exponent.

For example, the number 5 to the second power is the product of two factors, each of which is equal to 5. This product is equal to 25

Now imagine that we inadvertently multiplied base 5 by exponent 2

There was an error, because the number 5 to the second power is not equal to 10.

Additionally, it should be mentioned that the power of a number with an exponent of 1 is the number itself:

For example, the number 5 to the first power is the number 5 itself.

Accordingly, if the number does not have an indicator, then we must assume that the indicator is equal to one.

For example, the numbers 1, 2, 3 are given without an exponent, so their exponents will be equal to one. Each of these numbers can be written with an exponent of 1

And if you raise 0 to some power, you get 0. Indeed, no matter how many times nothing is multiplied by itself, nothing will turn out. Examples:

And the expression 0 0 makes no sense. But in some branches of mathematics, in particular analysis and set theory, the expression 0 0 can make sense.

For training, we will solve several examples of raising numbers to a power.

Example 1 Raise the number 3 to the second power.

The number 3 to the second power is the product of two factors, each of which is equal to 3

3 2 = 3 × 3 = 9

Example 2 Raise the number 2 to the fourth power.

The number 2 to the fourth power is the product of four factors, each of which is equal to 2

2 4 = 2 × 2 × 2 × 2 = 16

Example 3 Raise the number 2 to the third power.

The number 2 to the third power is the product of three factors, each of which is equal to 2

2 3 = 2 × 2 × 2 = 8

Exponentiation of the number 10

To raise the number 10 to a power, it is enough to add the number of zeros after the unit, equal to the exponent.

For example, let's raise the number 10 to the second power. First, we write the number 10 itself and indicate the number 2 as an indicator

10 2

Now we put an equal sign, write down one and after this one we write down two zeros, since the number of zeros should be equal to the exponent

10 2 = 100

So the number 10 to the second power is the number 100. This is due to the fact that the number 10 to the second power is the product of two factors, each of which is equal to 10

10 2 = 10 × 10 = 100

Example 2. Let's raise the number 10 to the third power.

In this case, there will be three zeros after the one:

10 3 = 1000

Example 3. Let's raise the number 10 to the fourth power.

In this case, there will be four zeros after the one:

10 4 = 10000

Example 4. Let's raise the number 10 to the first power.

In this case, there will be one zero after the one:

10 1 = 10

Representing the numbers 10, 100, 1000 as a power with base 10

To represent the numbers 10, 100, 1000, and 10000 as a power with base 10, you need to write base 10, and specify a number equal to the number of zeros in the original number as an exponent.

Let's represent the number 10 as a power with base 10. We see that it has one zero. So the number 10 as a power with base 10 will be represented as 10 1

10 = 10 1

Example 2. Let's represent the number 100 as a power with base 10. We see that the number 100 contains two zeros. So the number 100 as a power with base 10 will be represented as 10 2

100 = 10 2

Example 3. Let's represent the number 1000 as a power with base 10.

1 000 = 10 3

Example 4. Let's represent the number 10,000 as a power with base 10.

10 000 = 10 4

Exponentiation of a negative number

When raising a negative number to a power, it must be enclosed in parentheses.

For example, let's raise the negative number −2 to the second power. The number −2 to the second power is the product of two factors, each of which is equal to (−2)

(−2) 2 = (−2) × (−2) = 4

If we didn't parenthesize the number -2 , then it would turn out that we calculate the expression -2 2 , which not equal 4 . The expression -2² will be equal to -4 . To understand why, let's touch on some points.

When we put a minus in front of a positive number, we thereby perform the operation of taking the opposite value.

Let's say the number 2 is given, and you need to find its opposite number. We know that the opposite of 2 is −2. In other words, to find the opposite number for 2, it is enough to put a minus in front of this number. Inserting a minus in front of a number is already considered a full-fledged operation in mathematics. This operation, as mentioned above, is called the operation of taking the opposite value.

In the case of the expression -2 2, two operations occur: the operation of taking the opposite value and exponentiation. Raising to a power is a higher priority operation than taking the opposite value.

Therefore, the expression −2 2 is calculated in two steps. First, the exponentiation operation is performed. In this case, the positive number 2 was raised to the second power.

Then the opposite value was taken. This opposite value was found for the value 4. And the opposite value for 4 is −4

−2 2 = −4

Parentheses have the highest execution precedence. Therefore, in the case of calculating the expression (−2) 2, the opposite value is first taken, and then the negative number −2 is raised to the second power. The result is a positive answer of 4, since the product of negative numbers is a positive number.

Example 2. Raise the number −2 to the third power.

The number −2 to the third power is the product of three factors, each of which is equal to (−2)

(−2) 3 = (−2) × (−2) × (−2) = −8

Example 3. Raise the number −2 to the fourth power.

The number −2 to the fourth power is the product of four factors, each of which is equal to (−2)

(−2) 4 = (−2) × (−2) × (−2) × (−2) = 16

It is easy to see that when raising a negative number to a power, either a positive answer or a negative one can be obtained. The sign of the answer depends on the exponent of the initial degree.

If the exponent is even, then the answer is yes. If the exponent is odd, the answer is negative. Let's show this on the example of the number −3

In the first and third cases, the indicator was odd number, so the answer became negative.

In the second and fourth cases, the indicator was even number, so the answer became positive.

Example 7 Raise the number -5 to the third power.

The number -5 to the third power is the product of three factors, each of which is equal to -5. The exponent 3 is an odd number, so we can say in advance that the answer will be negative:

(−5) 3 = (−5) × (−5) × (−5) = −125

Example 8 Raise the number -4 to the fourth power.

The number -4 to the fourth power is the product of four factors, each of which is equal to -4. In this case, the indicator 4 is even, so we can say in advance that the answer will be positive:

(−4) 4 = (−4) × (−4) × (−4) × (−4) = 256

Finding Expression Values

When finding values ​​of expressions that do not contain brackets, exponentiation will be performed first, then multiplication and division in their order, and then addition and subtraction in their order.

Example 1. Find the value of the expression 2 + 5 2

First, exponentiation is performed. In this case, the number 5 is raised to the second power - it turns out 25. Then this result is added to the number 2

2 + 5 2 = 2 + 25 = 27

Example 10. Find the value of the expression −6 2 × (−12)

First, exponentiation is performed. Note that the number −6 is not in brackets, so the number 6 will be raised to the second power, then a minus will be placed in front of the result:

−6 2 × (−12) = −36 × (−12)

We complete the example by multiplying −36 by (−12)

−6 2 × (−12) = −36 × (−12) = 432

Example 11. Find the value of the expression −3 × 2 2

First, exponentiation is performed. Then the result is multiplied with the number −3

−3 × 2 2 = −3 × 4 = −12

If the expression contains brackets, then first you need to perform operations in these brackets, then exponentiation, then multiplication and division, and then addition and subtraction.

Example 12. Find the value of the expression (3 2 + 1 × 3) − 15 + 5

Let's do the parentheses first. Inside the brackets, we apply the previously learned rules, namely, first raise the number 3 to the second power, then perform the multiplication 1 × 3, then add the results of raising the number 3 to the power and multiplying 1 × 3. Then subtraction and addition are performed in the order in which they appear. Let's arrange the following order of performing the action on the original expression:

(3 2 + 1 × 3) - 15 + 5 = 12 - 15 + 5 = 2

Example 13. Find the value of the expression 2 × 5 3 + 5 × 2 3

First, we raise the numbers to a power, then we perform the multiplication and add the results:

2 x 5 3 + 5 x 2 3 = 2 x 125 + 5 x 8 = 250 + 40 = 290

Identity transformations of powers

Various identical transformations can be performed on powers, thereby simplifying them.

Suppose it was required to calculate the expression (2 3) 2 . In this example, two to the third power is raised to the second power. In other words, a degree is raised to another degree.

(2 3) 2 is the product of two powers, each of which is equal to 2 3

Moreover, each of these powers is the product of three factors, each of which is equal to 2

We got the product 2 × 2 × 2 × 2 × 2 × 2 , which is equal to 64. So the value of the expression (2 3) 2 or equal to 64

This example can be greatly simplified. For this, the indicators of the expression (2 3) 2 can be multiplied and this product can be written over the base 2

Got 2 6 . Two to the sixth power is the product of six factors, each of which is equal to 2. This product is equal to 64

This property works because 2 3 is the product of 2 × 2 × 2 , which in turn is repeated twice. Then it turns out that base 2 is repeated six times. From here we can write that 2 × 2 × 2 × 2 × 2 × 2 is 2 6

In general, for any reason a with indicators m And n, the following equality holds:

(a n)m = a n × m

This identical transformation is called exponentiation. It can be read like this: “When raising a power to a power, the base is left unchanged, and the exponents are multiplied” .

After multiplying the indicators, you get another degree, the value of which can be found.

Example 2. Find the value of the expression (3 2) 2

In this example, the base is 3, and the numbers 2 and 2 are the exponents. Let's use the rule of exponentiation. We leave the base unchanged, and multiply the indicators:

Got 3 4 . And the number 3 to the fourth power is 81

Let's look at the rest of the transformations.

Power multiplication

To multiply degrees, you need to separately calculate each degree, and multiply the results.

For example, let's multiply 2 2 by 3 3 .

2 2 is the number 4 and 3 3 is the number 27 . We multiply the numbers 4 and 27, we get 108

2 2 x 3 3 = 4 x 27 = 108

In this example, the bases of the powers were different. If the bases are the same, then one base can be written, and as an indicator, write the sum of the indicators of the initial degrees.

For example, multiply 2 2 by 2 3

In this example, the exponents have the same base. In this case, you can write one base 2 and write the sum of the exponents 2 2 and 2 3 as an indicator. In other words, leave the base unchanged, and add the exponents of the original degrees. It will look like this:

Got 2 5 . The number 2 to the fifth power is 32

This property works because 2 2 is the product of 2 × 2 and 2 3 is the product of 2 × 2 × 2 . Then the product of five identical factors is obtained, each of which is equal to 2. This product can be represented as 2 5

In general, for any a and indicators m And n the following equality holds:

This identical transformation is called the main property of the degree. It can be read like this: PWhen multiplying powers with the same base, the base is left unchanged, and the exponents are added. .

Note that this transformation can be applied to any number of degrees. The main thing is that the base is the same.

For example, let's find the value of the expression 2 1 × 2 2 × 2 3 . Foundation 2

In some problems, it may be sufficient to perform the corresponding transformation without calculating the final degree. This is of course very convenient, since it is not so easy to calculate large powers.

Example 1. Express as a power the expression 5 8 × 25

In this problem, you need to make it so that instead of the expression 5 8 × 25, one degree is obtained.

The number 25 can be represented as 5 2 . Then we get the following expression:

In this expression, you can apply the main property of the degree - leave the base 5 unchanged, and add the indicators 8 and 2:

Let's write the solution in short:

Example 2. Express as a power the expression 2 9 × 32

The number 32 can be represented as 2 5 . Then we get the expression 2 9 × 2 5 . Next, you can apply the base property of the degree - leave the base 2 unchanged, and add the indicators 9 and 5. This will result in the following solution:

Example 3. Calculate the 3 × 3 product using the basic power property.

Everyone is well aware that three times three is equal to nine, but the task requires using the main property of the degree in the course of solving. How to do it?

We recall that if a number is given without an indicator, then the indicator must be considered equal to one. So the factors 3 and 3 can be written as 3 1 and 3 1

3 1 × 3 1

Now we use the main property of the degree. We leave the base 3 unchanged, and add the indicators 1 and 1:

3 1 × 3 1 = 3 2 = 9

Example 4. Calculate the product 2 × 2 × 3 2 × 3 3 using the basic power property.

We replace the product 2 × 2 with 2 1 × 2 1 , then with 2 1 + 1 , and then with 2 2 . The product of 3 2 × 3 3 is replaced by 3 2 + 3 and then by 3 5

Example 5. Perform multiplication x × x

These are two identical alphabetic factors with indicators 1. For clarity, we write down these indicators. Further base x leave it unchanged, and add the indicators:

Being at the blackboard, one should not write down the multiplication of powers with the same bases in such detail as is done here. Such calculations must be done in the mind. A detailed entry will most likely annoy the teacher and he will lower the mark for this. Here, a detailed record is given so that the material is as accessible as possible for understanding.

The solution to this example should be written like this:

Example 6. Perform multiplication x 2 × x

The index of the second factor is equal to one. Let's write it down for clarity. Next, we leave the base unchanged, and add the indicators:

Example 7. Perform multiplication y 3 y 2 y

The index of the third factor is equal to one. Let's write it down for clarity. Next, we leave the base unchanged, and add the indicators:

Example 8. Perform multiplication aa 3 a 2 a 5

The index of the first factor is equal to one. Let's write it down for clarity. Next, we leave the base unchanged, and add the indicators:

Example 9. Express the power of 3 8 as a product of powers with the same base.

In this problem, you need to make a product of powers, the bases of which will be equal to 3, and the sum of the exponents of which will be equal to 8. You can use any indicators. We represent the degree 3 8 as the product of the powers 3 5 and 3 3

In this example, we again relied on the main property of the degree. After all, the expression 3 5 × 3 3 can be written as 3 5 + 3, whence 3 8 .

Of course, it was possible to represent the power 3 8 as a product of other powers. For example, in the form 3 7 × 3 1 , since this product is also 3 8

Representing a degree as a product of powers with the same base is mostly creative work. So don't be afraid to experiment.

Example 10. Submit Degree x 12 as various products of powers with bases x .

Let's use the main property of the degree. Imagine x 12 as products with bases x, and the sum of the exponents of which is equal to 12

The constructions with sums of indicators were recorded for clarity. Most of the time they can be skipped. Then we get a compact solution:

Exponentiation of a product

To raise a product to a power, you need to raise each factor of this product to the specified power and multiply the results.

For example, let's raise the product 2 × 3 to the second power. We take this product in brackets and indicate 2 as an indicator

Now let's raise each factor of the 2 × 3 product to the second power and multiply the results:

The principle of operation of this rule is based on the definition of the degree, which was given at the very beginning.

Raising the product of 2 × 3 to the second power means repeating this product twice. And if you repeat it twice, you can get the following:

2×3×2×3

From the permutation of the places of the factors, the product does not change. This allows you to group the same multipliers:

2×2×3×3

Repeating multipliers can be replaced with short entries - bases with exponents. The 2 × 2 product can be replaced by 2 2 , and the 3 × 3 product can be replaced by 3 2 . Then the expression 2 × 2 × 3 × 3 turns into the expression 2 2 × 3 2 .

Let be ab original work. To raise this product to the power n, you need to separately raise the factors a And b to the specified degree n

This property is valid for any number of factors. The following expressions are also valid:

Example 2. Find the value of the expression (2 × 3 × 4) 2

In this example, you need to raise the product 2 × 3 × 4 to the second power. To do this, you need to raise each factor of this product to the second power and multiply the results:

Example 3. Raise the product to the third power a×b×c

We enclose this product in brackets, and indicate the number 3 as an indicator

Example 4. Raise the product to the third power 3 xyz

We enclose this product in brackets, and indicate 3 as an indicator

(3xyz) 3

Let's raise each factor of this product to the third power:

(3xyz) 3 = 3 3 x 3 y 3 z 3

The number 3 to the third power is equal to the number 27. We leave the rest unchanged:

(3xyz) 3 = 3 3 x 3 y 3 z 3 = 27x 3 y 3 z 3

In some examples, the multiplication of powers with the same exponents can be replaced by the product of bases with the same exponent.

For example, let's calculate the value of the expression 5 2 × 3 2 . Raise each number to the second power and multiply the results:

5 2 x 3 2 = 25 x 9 = 225

But you can not calculate each degree separately. Instead, this product of powers can be replaced by a product with one exponent (5 × 3) 2 . Next, calculate the value in brackets and raise the result to the second power:

5 2 × 3 2 = (5 × 3) 2 = (15) 2 = 225

In this case, the rule of exponentiation of the product was again used. After all, if (a x b)n = a n × b n , then a n × b n = (a × b) n. That is, the left and right sides of the equation are reversed.

Exponentiation

We considered this transformation as an example when we tried to understand the essence of identical transformations of degrees.

When raising a power to a power, the base is left unchanged, and the exponents are multiplied:

(a n)m = a n × m

For example, the expression (2 3) 2 is raising a power to a power - two to the third power is raised to the second power. To find the value of this expression, the base can be left unchanged, and the exponents can be multiplied:

(2 3) 2 = 2 3 × 2 = 2 6

(2 3) 2 = 2 3 × 2 = 2 6 = 64

This rule is based on the previous rules: exponentiation of the product and the basic property of the degree.

Let's return to the expression (2 3) 2 . The expression in brackets 2 3 is the product of three identical factors, each of which is equal to 2. Then in the expression (2 3) 2 the power inside the brackets can be replaced by the product 2 × 2 × 2.

(2×2×2) 2

And this is the exponentiation of the product that we studied earlier. Recall that to raise a product to a power, you need to raise each factor of this product to the specified power and multiply the results:

(2 x 2 x 2) 2 = 2 2 x 2 2 x 2 2

Now we are dealing with the main property of the degree. We leave the base unchanged, and add the indicators:

(2 x 2 x 2) 2 = 2 2 x 2 2 x 2 2 = 2 2 + 2 + 2 = 2 6

As before, we got 2 6 . The value of this degree is 64

(2 x 2 x 2) 2 = 2 2 x 2 2 x 2 2 = 2 2 + 2 + 2 = 2 6 = 64

A product whose factors are also powers can also be raised to a power.

For example, let's find the value of the expression (2 2 × 3 2) 3 . Here, the indicators of each multiplier must be multiplied by the total indicator 3. Next, find the value of each degree and calculate the product:

(2 2 x 3 2) 3 = 2 2 x 3 x 3 2 x 3 = 2 6 x 3 6 = 64 x 729 = 46656

Approximately the same thing happens when raising to the power of a product. We said that when raising a product to a power, each factor of this product is raised to the indicated power.

For example, to raise the product of 2 × 4 to the third power, you need to write the following expression:

But earlier it was said that if a number is given without an indicator, then the indicator should be considered equal to one. It turns out that the factors of the product 2 × 4 initially have exponents equal to 1. This means that the expression 2 1 × 4 1 ​​was raised to the third power. And this is the raising of a degree to a power.

Let's rewrite the solution using the rule of exponentiation. We should get the same result:

Example 2. Find the value of the expression (3 3) 2

We leave the base unchanged, and multiply the indicators:

Got 3 6 . The number 3 to the sixth power is the number 729

Example 3xy)³

Example 4. Perform exponentiation in the expression ( abc)⁵

Let's raise each factor of the product to the fifth power:

Example 5ax) 3

Let's raise each factor of the product to the third power:

Since the negative number −2 was raised to the third power, it was taken in brackets.

Example 6. Perform exponentiation in expression (10 xy) 2

Example 7. Perform exponentiation in the expression (−5 x) 3

Example 8. Perform exponentiation in the expression (−3 y) 4

Example 9. Perform exponentiation in the expression (−2 abx)⁴

Example 10. Simplify the expression x 5×( x 2) 3

Degree x 5 will remain unchanged for now, and in the expression ( x 2) 3 perform the exponentiation to the power:

x 5 × (x 2) 3 = x 5 × x 2×3 = x 5 × x 6

Now let's do the multiplication x 5 × x 6. To do this, we use the main property of the degree - the base x leave it unchanged, and add the indicators:

x 5 × (x 2) 3 = x 5 × x 2×3 = x 5 × x 6 = x 5 + 6 = x 11

Example 9. Find the value of the expression 4 3 × 2 2 using the basic property of the degree.

The main property of the degree can be used if the bases of the initial degrees are the same. In this example, the bases are different, therefore, to begin with, the original expression needs to be slightly modified, namely, to make the bases of the degrees become the same.

Let's look closely at the power of 4 3 . The base of this degree is the number 4, which can be represented as 2 2 . Then the original expression will take the form (2 2) 3 × 2 2 . By exponentiating to a power in the expression (2 2) 3 , we get 2 6 . Then the original expression will take the form 2 6 × 2 2 , which can be calculated using the main property of the degree.

Let's write the solution of this example:

Division of powers

To perform power division, you need to find the value of each power, then perform the division of ordinary numbers.

For example, let's divide 4 3 by 2 2 .

Calculate 4 3 , we get 64 . We calculate 2 2 , we get 4. Now we divide 64 by 4, we get 16

If, when dividing the degrees of the base, they turn out to be the same, then the base can be left unchanged, and the exponent of the divisor can be subtracted from the exponent of the dividend.

For example, let's find the value of the expression 2 3: 2 2

We leave the base 2 unchanged, and subtract the exponent of the divisor from the exponent of the dividend:

So the value of the expression 2 3: 2 2 is 2 .

This property is based on the multiplication of powers with the same bases, or, as we used to say, on the main property of the degree.

Let's return to the previous example 2 3: 2 2 . Here the dividend is 2 3 and the divisor is 2 2 .

To divide one number by another means to find a number that, when multiplied by a divisor, will give the dividend as a result.

In our case, dividing 2 3 by 2 2 means finding a power that, when multiplied by the divisor 2 2, will result in 2 3 . What power can be multiplied by 2 2 to get 2 3 ? Obviously, only the degree 2 1 . From the main property of the degree we have:

You can verify that the value of the expression 2 3: 2 2 is 2 1 by directly evaluating the expression 2 3: 2 2 . To do this, first we find the value of the degree 2 3 , we get 8 . Then we find the value of the degree 2 2 , we get 4 . Divide 8 by 4, we get 2 or 2 1 , since 2 = 2 1 .

2 3: 2 2 = 8: 4 = 2

Thus, when dividing powers with the same base, the following equality holds:

It may also happen that not only the bases, but also the indicators may be the same. In this case, the answer will be one.

For example, let's find the value of the expression 2 2: 2 2 . Let's calculate the value of each degree and perform the division of the resulting numbers:

When solving example 2 2: 2 2, you can also apply the rule for dividing degrees with the same bases. The result is a number to the zero power, since the difference between the exponents of 2 2 and 2 2 is zero:

Why the number 2 to the zero degree is equal to one, we found out above. If you calculate 2 2: 2 2 in the usual way, without using the rule for dividing degrees, you get one.

Example 2. Find the value of the expression 4 12: 4 10

We leave 4 unchanged, and subtract the exponent of the divisor from the exponent of the dividend:

4 12: 4 10 = 4 12 − 10 = 4 2 = 16

Example 3. Submit private x 3: x as a degree with a base x

Let's use the rule of division of powers. Base x leave it unchanged, and subtract the exponent of the divisor from the exponent of the dividend. The divisor exponent is equal to one. For clarity, let's write it down:

Example 4. Submit private x 3: x 2 as a power with a base x

Let's use the rule of division of powers. Base x

The division of degrees can be written as a fraction. So, the previous example can be written as follows:

The numerator and denominator of a fraction can be written in expanded form, namely in the form of products of identical factors. Degree x 3 can be written as x × x × x, and the degree x 2 as x × x. Then the construction x 3 − 2 can be skipped and use fraction reduction. In the numerator and in the denominator, it will be possible to reduce two factors each x. The result will be one multiplier x

Or even shorter:

Also, it is useful to be able to quickly reduce fractions consisting of powers. For example, a fraction can be reduced to x 2. To reduce a fraction by x 2 you need to divide the numerator and denominator of the fraction by x 2

The division of degrees can not be described in detail. The above abbreviation can be made shorter:

Or even shorter:

Example 5. Execute division x 12 : x 3

Let's use the rule of division of powers. Base x leave it unchanged, and subtract the exponent of the divisor from the exponent of the dividend:

We write the solution using fraction reduction. Division of degrees x 12 : x 3 will be written as . Next, we reduce this fraction by x 3 .

Example 6. Find the value of an expression

In the numerator, we perform the multiplication of powers with the same bases:

Now we apply the rule for dividing powers with the same bases. We leave the base 7 unchanged, and subtract the exponent of the divisor from the exponent of the dividend:

We complete the example by calculating the power of 7 2

Example 7. Find the value of an expression

Let's perform exponentiation in the numerator. You need to do this with the expression (2 3) 4

Now let's perform the multiplication of powers with the same bases in the numerator.