Formula for raising to the 5th power. Degree and its properties

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

Number c is n-th power of a number a When:

Operations with degrees.

1. By multiplying degrees with the same base, their indicators are added:

a m·a n = a m + n .

2. When dividing degrees with the same base, their exponents 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:

(a m) n = a m n .

Each formula above is true 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 a 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 radical number to this power:

4. If you increase the degree of the root in n once and at the same time build into n th power is a radical number, then the value of the root will not change:

5. If you reduce the degree of the root in n extract the root at the same time n-th power of a radical number, then the value of the root will not change:

A degree with a negative exponent. The power of a certain number with a non-positive (integer) exponent is defined as one divided by the power 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 with m< n.

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

To formula a m:a n =a m - n became fair when m=n, the presence of zero degree is required.

A degree with a zero index. The power of any number not equal to zero with a zero exponent is equal to one.

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

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

can be found using multiplication. For example: 5+5+5+5+5+5=5x6. Such an expression is said to be that the sum of equal terms is folded into a product. And vice versa, if we read this equality from right to left, we find that we have expanded the sum of equal terms. Similarly, you can collapse the product of several equal factors 5x5x5x5x5x5=5 6.

That is, instead of multiplying six identical factors 5x5x5x5x5x5, they write 5 6 and say “five to the sixth power.”

The expression 5 6 is a power of a number, where:

5 - degree base;

6 - exponent.

Actions by which the product of equal factors is reduced to a power are called raising to a power.

In general, a degree with base “a” and exponent “n” is written as follows

Raising the number a to the power n means finding the product of n factors, each of which is equal to a

If the base of the degree “a” is equal to 1, then the value of the degree for any natural n will be equal to 1. For example, 1 5 =1, 1 256 =1

If you raise the number “a” to first degree, then we get the number a itself: a 1 = a

If you raise any number to zero degree, then as a result of calculations we get one. a 0 = 1

The second and third powers of a number are considered special. They came up with names for them: the second degree is called square the number, third - cube this number.

Any number can be raised to a power - positive, negative or zero. In this case, the following rules do not apply:

When finding the power of a positive number, the result is a positive number.

When calculating zero to the natural power, we get zero.

x m · x n = x m + n

for example: 7 1.7 7 - 0.9 = 7 1.7+(- 0.9) = 7 1.7 - 0.9 = 7 0.8

To divide powers with the same bases We do not change the base, but subtract the exponents:

x m / x n = x m - n , Where, m > n,

for example: 13 3.8 / 13 -0.2 = 13 (3.8 -0.2) = 13 3.6

When calculating raising a power to a power We do not change the base, but multiply the exponents by each other.

(at m ) n = y m n

for example: (2 3) 2 = 2 3 2 = 2 6

(X · y) n = x n · y m ,

for example:(2 3) 3 = 2 n 3 m,

When performing calculations according to raising a fraction to a power we raise the numerator and denominator of the fraction to a given power

(x/y)n = x n / y n

for example: (2 / 5) 3 = (2 / 5) · (2 ​​/ 5) · (2 ​​/ 5) = 2 3 / 5 3.

The sequence of calculations when working with expressions containing a degree.

When performing calculations of expressions without parentheses, but containing powers, first of all, they perform exponentiation, then multiplication and division, and only then addition and subtraction operations.

If you need to calculate an expression containing brackets, then first do the calculations in the brackets in the order indicated above, and then the remaining actions in the same order from left to right.

Very widely in practical calculations, ready-made tables of powers are used to simplify calculations.

If we ignore the eighth power, what do we see here? Let's remember the 7th grade program. So, do you remember? This is the formula for abbreviated multiplication, namely the difference of squares! We get:

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

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

Magically the terms changed places. This “phenomenon” applies to any expression to an even degree: we can easily change the signs in parentheses.

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

Let's go back to the example:

And again the formula:

Whole we call the natural numbers, 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, let us ask ourselves: why is this so?

Let's consider some degree with a base. Take, for example, and multiply by:

So, we multiplied the number by, and we got the same thing as it was - . What number should you multiply 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 will still get zero, this is clear. But on the other hand, like any number to the zero power, it must be equal. So how much of this is true? The mathematicians decided not to get involved and refused to raise zero to the zero power. That is, now we cannot not only divide by zero, but also raise it to the zero power.

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

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

Now let’s extend the resulting rule to an arbitrary degree:

So, let's formulate a rule:

A number with a negative power is the reciprocal of the same number with a positive power. But at the same time The base cannot be null:(because you can’t divide by).

Let's summarize:

I. The expression is not defined in the case. If, then.

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

III. A number 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 independent solutions:

Analysis of problems for independent solution:

I know, I know, the numbers are scary, but on the Unified State Exam you have to be prepared for anything! Solve these examples or analyze their solutions if you couldn’t solve them and you will learn to cope with them easily in the exam!

Let's continue to expand the range of numbers “suitable” as an exponent.

Now let's consider rational numbers. What numbers are called rational?

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

To understand what it is "fractional degree", consider the fraction:

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

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

What number must be raised to a power to get?

This formulation is the definition of the 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 to.

That is, the root of the th power is the inverse operation of raising to a power: .

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

Now we add the numerator: what is it? The answer is easy to obtain using the power-to-power rule:

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

None!

Let us remember the rule: any number raised to an even power is a positive number. That is, it is impossible to extract even roots from negative numbers!

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 the expression?

But here a problem arises.

The number can be represented in the form of other, reducible fractions, for example, or.

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

Or another example: once, then you can write it down. But if we write down the indicator differently, we will again get into trouble: (that is, we got a completely different result!).

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

So if:

• — natural number;
• - integer;

Examples:

Rational exponents are very useful for transforming expressions with roots, for example:

5 examples to practice

Analysis of 5 examples for training

1. Don't forget about the usual properties of degrees:

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

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

Well, now comes the hardest part. Now we'll figure it out degree with irrational exponent.

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

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

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

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

...number to the zeroth power- this is, as it were, a number multiplied by itself once, that is, they have not yet begun to multiply it, which means that the number itself has not even appeared yet - therefore the result is only a certain “blank number”, namely a number;

...negative integer degree- it’s as if some “reverse process” had occurred, 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, the 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 to solve such examples :))

For example:

Decide for yourself:

Analysis of solutions:

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

Now look at the indicator. Doesn't he remind you of anything? Let us recall the formula for abbreviated multiplication of difference of squares:

In this case,

It turns out that:

Answer: .

2. We reduce fractions in exponents to the same form: either both decimals or both ordinary ones. We get, for example:

Answer: 16

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

ADVANCED LEVEL

Determination of degree

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

• — degree base;
• - exponent.

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

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

Degree with an integer exponent (0, ±1, ±2,...)

If the exponent is positive integer number:

Construction to the zero degree:

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 negative integer number:

(because you can’t divide by).

Once again about zeros: the expression is not defined in the case. If, then.

Examples:

Power with rational exponent

• — natural number;
• - integer;

Examples:

Properties of degrees

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 we get the following product:

But by definition it 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 there must be the same reasons. Therefore, we combine the powers with the base, but it remains a separate factor:

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

Under no circumstances can you write that.

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

Let's regroup this work like this:

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

In essence, this can be called “taking the indicator out of brackets.” But you can never do this in total: !

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

Power with a negative base.

Up to this point we have only discussed what it should be like indicator degrees. But what should be the basis? In powers of natural indicator the basis may be any number .

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

For example, is the number positive or negative? A? ?

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

But the negative ones are a little more interesting. We remember the simple rule from 6th grade: “minus for 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. The following simple rules can be formulated:

1. even degree, - number positive.
2. Negative number raised to odd degree, - number negative.
3. A positive number to any degree 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: after all, it doesn’t matter what the base is equal to - the degree is even, which means the result will always be positive. Well, except when the base is zero. The base is not equal, is it? Obviously not, since (because).

Example 6) is no longer so simple. Here you need to find out which is less: or? If we remember that, it becomes clear that, which means 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 by each other, divide them into pairs and get:

Before we look at the last rule, let's solve a few examples.

Calculate the expressions:

Solutions :

If we ignore the eighth power, what do we see here? Let's remember the 7th grade program. So, do you remember? This is the formula for abbreviated multiplication, namely the difference of squares!

We get:

Let's look carefully at the denominator. It looks a lot like one of the numerator factors, but what's wrong? The order of the terms is wrong. If they were reversed, Rule 3 could apply. But how? 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 turns out like this:

Magically the terms changed places. This “phenomenon” applies to any expression to an even degree: we can easily change the signs in parentheses. But it's important to remember: All signs change at the same time! You can’t replace it with by changing only one disadvantage we don’t like!

Let's go back to the example:

And again the formula:

So now the last rule:

How will we prove it? Of course, as usual: let’s expand on the concept of degree and simplify it:

Well, now let's open the brackets. How many letters are there in total? times by multipliers - what does this remind you of? This is nothing more than a definition of the operation multiplication: There were only multipliers there. That is, this, by definition, is a power of a number with an exponent:

Example:

Degree with irrational exponent

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

When studying degrees with natural, integer and rational exponents, each time we created a certain “image”, “analogy”, or description in more familiar terms. For example, a degree with a natural exponent is a number multiplied by itself several times; a number to the zero power is, as it were, a number multiplied by itself once, that is, they have not yet begun to multiply it, which means that the number itself has not even appeared yet - therefore the result is only a certain “blank number”, namely a number; a degree with an integer negative exponent - it’s as if some “reverse process” had 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). It is rather a purely mathematical object that mathematicians created to extend the concept of degree to the entire space of numbers.

By the way, in science a degree with a complex exponent is often used, that is, the 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. Let's remember the difference of squares formula. Answer: .
2. We reduce the fractions to the same form: either both decimals or both ordinary ones. We get, for example: .
3. Nothing special, we use the usual properties of degrees:

SUMMARY OF THE SECTION AND BASIC FORMULAS

Degree called an expression of the form: , where:

Degree with an integer exponent

a degree whose exponent is a natural number (i.e., integer and positive).

Power with rational exponent

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

Degree with irrational exponent

a degree whose exponent is an infinite decimal fraction or root.

Properties of degrees

Features of degrees.

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

NOW YOU HAVE THE WORD...

How do you like the article? Write below in the comments whether you liked it or not.

Tell us about your experience using degree properties.

Perhaps you have questions. Or suggestions.

Write in the comments.

And good luck on your exams!

The calculator helps you quickly raise a number to a power online. The base of the degree can be any number (both integers and reals). The exponent can also be an integer or real, and can also be positive or negative. Please remember that for negative numbers, raising to a non-integer power is undefined and therefore the calculator will report an error if you attempt it.

Degree calculator

Raise to power

Exponentiations: 94722

What is a natural power of a number?

The number p is called the nth power of a number if p is equal to the number a multiplied by itself n times: p = a n = a·...·a
n - called exponent, and the number a is degree basis.

How to raise a number to a natural power?

To understand how to raise various numbers to natural powers, consider a few examples:

Example 1. Raise the number three to the fourth power. That is, it is necessary to calculate 3 4
Solution: as mentioned above, 3 4 = 3·3·3·3 = 81.
Answer: 3 4 = 81 .

Example 2. Raise the number five to the fifth power. That is, it is necessary to calculate 5 5
Solution: similarly, 5 5 = 5·5·5·5·5 = 3125.
Answer: 5 5 = 3125 .

Thus, to raise a number to a natural power, you just need to multiply it by itself n times.

What is a negative power of a number?

The negative power -n of a is one divided by a to the power of n: a -n = .

In this case, a negative power exists only for non-zero numbers, since otherwise division by zero would occur.

How to raise a number to a negative integer power?

To raise a non-zero number to a negative power, you need to calculate the value of this number to the same positive power and divide one by the result.

Example 1. Raise the number two to the negative fourth power. That is, you need to calculate 2 -4

Solution: as stated above, 2 -4 = = = 0.0625.

Answer: 2 -4 = 0.0625 .

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

The concept of exponentiation

Let's start by formulating basic definitions.

Definition 1

Exponentiation- this is the calculation of the value of the power of a certain number.

That is, the words “calculating the value of a power” and “raising to a power” mean the same thing. So, if the problem says “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 present the basic rules that must be followed when making such calculations.

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

Example 1

Condition: raise - 2 to the power 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 . Previously, we looked at how to correctly multiply the mixed numbers mentioned in the condition.

Let's perform these steps and get the answer: 3 2 7 · 3 2 7 = 23 7 · 23 7 = 529 49 = 10 39 49

If the problem indicates the need to raise irrational numbers to a natural power, we will need to first round their bases to the digit that will allow us to obtain an answer of the required accuracy. Let's look at an example.

Example 3

Perform the square of π.

Solution

First, let's round it to the nearest hundredth. Then π 2 ≈ (3, 14) 2 = 9, 8596. If π ≈ 3. 14159, then we get a more accurate result: π 2 ≈ (3, 14159) 2 = 9, 8695877281.

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

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

This is clear from the recording .

It does not depend on the basis of the degree.

Example 4

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

For convenience, we will examine 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 talked above about how to work with such degrees.

Now let's see how to correctly raise to the zero power. For a base other than zero, this calculation always outputs 1. We 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 an integer negative 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 power with a positive integer exponent, and we have already learned how to calculate it. Let's give examples of tasks.

Example 6

Raise 2 to the power - 3.

Solution

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

Let's calculate the denominator of this fraction and get 8: 2 3 = 2 · 2 · 2 = 8.

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

Example 7

Raise 1.43 to the -2 power.

Solution

Let's 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. All we have to do is write this result in the form of an ordinary fraction, for which we need to multiply it by 10 thousand (see the material on converting fractions).

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

A special case is raising a number to the minus first power. The value of this degree is equal to the reciprocal of the original value of the base: a - 1 = 1 a 1 = 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 remember the basic definition of a degree with a fractional exponent: a m n = a m n for any positive a, integer m and natural n.

Definition 2

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

We have the equality a m n = a m n , which, taking into account 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 a number a to a fractional power m / n, then first we take the nth root of 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 = 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. Transform the basic equality: 8 - 2 3 = 8 - 2 3 = 8 3 - 2

After this, 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 it any way you like.

There are cases when the degree has an indicator expressed as a mixed number or a decimal fraction. To simplify calculations, it is better to replace it with an ordinary fraction and calculate 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.

Now we carry out in order all the actions indicated above: 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: 13 501, 25107.

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

Let us separately dwell on powers 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 a power whose exponent 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 a power 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 is, the more accurate the answer. Let's show with an example:

Example 11

Calculate the approximation of 2 to the power of 1.174367....

Solution

Let us limit ourselves to the decimal approximation a n = 1, 17. Let's carry out 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 accurate: 2 1, 174367. . . ≈ 2 1, 1743 ≈ 2, 256833.

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