In this material we will look at what a power of a number is. In addition to the basic definitions, we will formulate what powers with natural, integer, rational and irrational exponents are. As always, all concepts will be illustrated with example problems.

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First, let's formulate the basic definition of a degree with a natural exponent. To do this, we need to remember the basic rules of multiplication. Let us clarify in advance that for now we will take a real number as a base (denoted by the letter a), and a natural number as an indicator (denoted by the letter n).

Definition 1

The power of a number a with natural exponent n is the product of the nth number of factors, each of which is equal to the number a. The degree is written like this: a n, and in the form of a formula its composition can be represented as follows:

For example, if the exponent is 1 and the base is a, then the first power of a is written as a 1. Given that a is the value of the factor and 1 is the number of factors, we can conclude that a 1 = a.

In general, we can say that a degree is a convenient form of writing a large number of equal factors. So, a record of the form 8 8 8 8 can be shortened to 8 4 . In much the same way, the product helps us avoid writing a large number of terms (8 + 8 + 8 + 8 = 8 · 4); We have already discussed this in the article devoted to the multiplication of natural numbers.

How to correctly read the degree entry? The generally accepted option is “a to the power of n”. Or you can say “nth power of a” or “anth power”. If, say, in the example we encountered the entry 8 12 , we can read "8 to the 12th power", "8 to the power of 12" or "12th power of 8".

The second and third powers of numbers have their own established names: square and cube. If we see the second power, for example, the number 7 (7 2), then we can say “7 squared” or “square of the number 7”. Similarly, the third degree is read like this: 5 3 - this is the “cube of the number 5” or “5 cubed.” However, you can also use the standard formulation “to the second/third power”; this will not be a mistake.

Example 1

Let's look at an example of a degree with a natural exponent: for 5 7 five will be the base, and seven will be the exponent.

The base does not have to be an integer: for the degree (4 , 32) 9 the base will be the fraction 4, 32, and the exponent will be nine. Pay attention to the parentheses: this notation is made for all powers whose bases differ from natural numbers.

For example: 1 2 3, (- 3) 12, - 2 3 5 2, 2, 4 35 5, 7 3.

What are parentheses for? They help avoid errors in calculations. Let's say we have two entries: (− 2) 3 And − 2 3 . The first of these means a negative number minus two raised to a power with a natural exponent of three; the second is the number corresponding to the opposite value of the degree 2 3 .

Sometimes in books you can find a slightly different spelling of the power of a number - a^n(where a is the base and n is the exponent). That is, 4^9 is the same as 4 9 . If n is a multi-digit number, it is placed in parentheses. For example, 15 ^ (21) , (− 3 , 1) ^ (156) . But we will use the notation a n as more common.

It’s easy to guess how to calculate the value of an exponent with a natural exponent from its definition: you just need to multiply a nth number of times. We wrote more about this in another article.

The concept of degree is the inverse of another mathematical concept - the root of a number. If we know the value of the power and the exponent, we can calculate its base. The degree has some specific properties that are useful for solving problems, which we discussed in a separate material.

Exponents can include not only natural numbers, but also any integer values ​​in general, including negative ones and zeros, because they also belong to the set of integers.

Definition 2

The power of a number with a positive integer exponent can be represented as a formula: .

In this case, n is any positive integer.

Let's understand the concept of zero degree. To do this, we use an approach that takes into account the quotient property for powers with equal bases. It is formulated like this:

Definition 3

Equality a m: a n = a m − n will be true under the following conditions: m and n are natural numbers, m< n , a ≠ 0 .

The last condition is important because it avoids division by zero. If the values ​​of m and n are equal, then we get the following result: a n: a n = a n − n = a 0

But at the same time a n: a n = 1 is the quotient of equal numbers a n and a. It turns out that the zero power of any non-zero number is equal to one.

However, such a proof does not apply to zero to the zeroth power. To do this, we need another property of powers - the property of products of powers with equal bases. It looks like this: a m · a n = a m + n .

If n is equal to 0, then a m · a 0 = a m(this equality also proves to us that a 0 = 1). But if and is also equal to zero, our equality takes the form 0 m · 0 0 = 0 m, It will be true for any natural value of n, and it does not matter what exactly the value of the degree is equal to 0 0 , that is, it can be equal to any number, and this will not affect the accuracy of the equality. Therefore, a notation of the form 0 0 does not have its own special meaning, and we will not attribute it to it.

If desired, it is easy to check that a 0 = 1 converges with the degree property (a m) n = a m n provided that the base of the degree is not zero. Thus, the power of any non-zero number with exponent zero is one.

Example 2

Let's look at an example with specific numbers: So, 5 0 - unit, (33 , 3) 0 = 1 , - 4 5 9 0 = 1 , and the value 0 0 undefined.

After the zero degree, we just have to figure out what a negative degree is. To do this, we need the same property of the product of powers with equal bases that we already used above: a m · a n = a m + n.

Let us introduce the condition: m = − n, then a should not be equal to zero. It follows that a − n · a n = a − n + n = a 0 = 1. It turns out that a n and a−n we have mutually reciprocal numbers.

As a result, a to the negative whole power is nothing more than the fraction 1 a n.

This formulation confirms that for a degree with an integer negative exponent, all the same properties are valid that a degree with a natural exponent has (provided that the base is not equal to zero).

Example 3

A power a with a negative integer exponent n can be represented as a fraction 1 a n . Thus, a - n = 1 a n subject to a ≠ 0 and n is any natural number.

Let us illustrate our idea with specific examples:

Example 4

3 - 2 = 1 3 2 , (- 4 . 2) - 5 = 1 (- 4 . 2) 5 , 11 37 - 1 = 1 11 37 1

In the last part of the paragraph, we will try to depict everything that has been said clearly in one formula:

Definition 4

The power of a number with a natural exponent z is: a z = a z, e with l and z - positive integer 1, z = 0 and a ≠ 0, (for z = 0 and a = 0 the result is 0 0, the values ​​of the expression 0 0 are not is defined) 1 a z, if and z is a negative integer and a ≠ 0 ( if z is a negative integer and a = 0 you get 0 z, egoz the value is undetermined)

What are powers with a rational exponent?

We examined cases when the exponent contains an integer. However, you can raise a number to a power even when its exponent contains a fractional number. This is called a power with a rational exponent. In this section we will prove that it has the same properties as other powers.

What are rational numbers? Their set includes both whole and fractional numbers, and fractional numbers can be represented as ordinary fractions (both positive and negative). Let us formulate the definition of the power of a number a with a fractional exponent m / n, where n is a natural number and m is an integer.

We have some degree with a fractional exponent a m n . In order for the power to power property to hold, the equality a m n n = a m n · n = a m must be true.

Given the definition of the nth root and that a m n n = a m, we can accept the condition a m n = a m n if a m n makes sense for the given values ​​of m, n and a.

The above properties of a degree with an integer exponent will be true under the condition a m n = a m n .

The main conclusion from our reasoning is this: the power of a certain number a with a fractional exponent m / n is the nth root of the number a to the power m. This is true if, for given values ​​of m, n and a, the expression a m n remains meaningful.

1. We can limit the value of the base of the degree: let's take a, which for positive values ​​of m will be greater than or equal to 0, and for negative values ​​- strictly less (since for m ≤ 0 we get 0 m, but such a degree is not defined). In this case, the definition of a degree with a fractional exponent will look like this:

A power with a fractional exponent m/n for some positive number a is the nth root of a raised to the power m. This can be expressed as a formula:

For a power with a zero base, this provision is also suitable, but only if its exponent is a positive number.

A power with a base zero and a fractional positive exponent m/n can be expressed as

0 m n = 0 m n = 0 provided m is a positive integer and n is a natural number.

For a negative ratio m n< 0 степень не определяется, т.е. такая запись смысла не имеет.

Let's note one point. Since we introduced the condition that a is greater than or equal to zero, we ended up discarding some cases.

The expression a m n sometimes still makes sense for some negative values ​​of a and some m. Thus, the correct entries are (- 5) 2 3, (- 1, 2) 5 7, - 1 2 - 8 4, in which the base is negative.

2. The second approach is to consider separately the root a m n with even and odd exponents. Then we will need to introduce one more condition: the degree a, in the exponent of which there is a reducible ordinary fraction, is considered to be the degree a, in the exponent of which there is the corresponding irreducible fraction. Later we will explain why we need this condition and why it is so important. Thus, if we have the notation a m · k n · k , then we can reduce it to a m n and simplify the calculations.

If n is an odd number and the value of m is positive and a is any non-negative number, then a m n makes sense. The condition for a to be non-negative is necessary because a root of an even degree cannot be extracted from a negative number. If the value of m is positive, then a can be both negative and zero, because The odd root can be taken from any real number.

Let's combine all the above definitions in one entry:

Here m/n means an irreducible fraction, m is any integer, and n is any natural number.

Definition 5

For any ordinary reducible fraction m · k n · k the degree can be replaced by a m n .

The power of a number a with an irreducible fractional exponent m / n – can be expressed as a m n in the following cases: - for any real a, positive integer values ​​m and odd natural values ​​n. Example: 2 5 3 = 2 5 3, (- 5, 1) 2 7 = (- 5, 1) - 2 7, 0 5 19 = 0 5 19.

For any non-zero real a, negative integer values ​​of m and odd values ​​of n, for example, 2 - 5 3 = 2 - 5 3, (- 5, 1) - 2 7 = (- 5, 1) - 2 7

For any non-negative a, positive integer m and even n, for example, 2 1 4 = 2 1 4, (5, 1) 3 2 = (5, 1) 3, 0 7 18 = 0 7 18.

For any positive a, negative integer m and even n, for example, 2 - 1 4 = 2 - 1 4, (5, 1) - 3 2 = (5, 1) - 3, .

In the case of other values, the degree with a fractional exponent is not determined. Examples of such degrees: - 2 11 6, - 2 1 2 3 2, 0 - 2 5.

Now let’s explain the importance of the condition discussed above: why replace a fraction with a reducible exponent with a fraction with an irreducible exponent. If we had not done this, we would have had the following situations, say, 6/10 = 3/5. Then it should be true (- 1) 6 10 = - 1 3 5 , but - 1 6 10 = (- 1) 6 10 = 1 10 = 1 10 10 = 1 , and (- 1) 3 5 = (- 1) 3 5 = - 1 5 = - 1 5 5 = - 1 .

The definition of a degree with a fractional exponent, which we presented first, is more convenient to use in practice than the second, so we will continue to use it.

Definition 6

Thus, the power of a positive number a with a fractional exponent m/n is defined as 0 m n = 0 m n = 0. In case of negative a the notation a m n does not make sense. Power of zero for positive fractional exponents m/n is defined as 0 m n = 0 m n = 0 , for negative fractional exponents we do not define the degree of zero.

In conclusions, we note that you can write any fractional indicator both as a mixed number and as a decimal fraction: 5 1, 7, 3 2 5 - 2 3 7.

When calculating, it is better to replace the exponent with an ordinary fraction and then use the definition of exponent with a fractional exponent. For the examples above we get:

5 1 , 7 = 5 17 10 = 5 7 10 3 2 5 - 2 3 7 = 3 2 5 - 17 7 = 3 2 5 - 17 7

What are powers with irrational and real exponents?

What are real numbers? Their set includes both rational and irrational numbers. Therefore, in order to understand what a degree with a real exponent is, we need to define degrees with rational and irrational exponents. We have already mentioned rational ones above. Let's deal with irrational indicators step by step.

Example 5

Let's assume that we have an irrational number a and a sequence of its decimal approximations a 0 , a 1 , a 2 , . . . . For example, let's take the value a = 1.67175331. . . , Then

a 0 = 1, 6, a 1 = 1, 67, a 2 = 1, 671, . . . , a 0 = 1.67, a 1 = 1.6717, a 2 = 1.671753, . . .

We can associate sequences of approximations with a sequence of degrees a a 0 , a a 1 , a a 2 , . . . . If we remember what we said earlier about raising numbers to rational powers, then we can calculate the values ​​of these powers ourselves.

Let's take for example a = 3, then a a 0 = 3 1, 67, a a 1 = 3 1, 6717, a a 2 = 3 1, 671753, . . . etc.

The sequence of powers can be reduced to a number, which will be the value of the power with base a and irrational exponent a. As a result: a degree with an irrational exponent of the form 3 1, 67175331. . can be reduced to the number 6, 27.

Definition 7

The power of a positive number a with an irrational exponent a is written as a a . Its value is the limit of the sequence a a 0 , a a 1 , a a 2 , . . . , where a 0 , a 1 , a 2 , . . . are successive decimal approximations of the irrational number a. A degree with a zero base can also be defined for positive irrational exponents, with 0 a = 0 So, 0 6 = 0, 0 21 3 3 = 0. But this cannot be done for negative ones, since, for example, the value 0 - 5, 0 - 2 π is not defined. A unit raised to any irrational power remains a unit, for example, and 1 2, 1 5 in 2 and 1 - 5 will be equal to 1.

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In this article we will figure out what it is degree of. Here we will give definitions of the power of a number, while we will consider in detail all possible exponents, starting with the natural exponent and ending with the irrational one. In the material you will find a lot of examples of degrees, covering all the subtleties that arise.

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Power with natural exponent, square of a number, cube of a number

Let's start with . Looking ahead, let's say that the definition of the power of a number a with natural exponent n is given for a, which we will call degree basis, and n, which we will call exponent. We also note that a degree with a natural exponent is determined through a product, so to understand the material below you need to have an understanding of multiplying numbers.

Definition.

Power of a number with natural exponent n is an expression of the form a n, the value of which is equal to the product of n factors, each of which is equal to a, that is, .
In particular, the power of a number a with exponent 1 is the number a itself, that is, a 1 =a.

It’s worth mentioning right away about the rules for reading degrees. The universal way to read the notation a n is: “a to the power of n”. In some cases, the following options are also acceptable: “a to the nth power” and “nth power of a”. For example, let's take the power 8 12, this is “eight to the power of twelve”, or “eight to the twelfth power”, or “twelfth power of eight”.

The second power of a number, as well as the third power of a number, have their own names. The second power of a number is called square the number, for example, 7 2 is read as “seven squared” or “the square of the number seven.” The third power of a number is called cubed numbers, for example, 5 3 can be read as “five cubed” or you can say “cube of the number 5”.

It's time to bring examples of degrees with natural exponents. Let's start with the degree 5 7, here 5 is the base of the degree, and 7 is the exponent. Let's give another example: 4.32 is the base, and the natural number 9 is the exponent (4.32) 9 .

Please note that in the last example, the base of the power 4.32 is written in parentheses: to avoid discrepancies, we will put in parentheses all bases of the power that are different from natural numbers. As an example, we give the following degrees with natural exponents , their bases are not natural numbers, so they are written in parentheses. Well, for complete clarity, at this point we will show the difference contained in records of the form (−2) 3 and −2 3. The expression (−2) 3 is a power of −2 with a natural exponent of 3, and the expression −2 3 (it can be written as −(2 3) ) corresponds to the number, the value of the power 2 3 .

Note that there is a notation for the power of a number a with an exponent n of the form a^n. Moreover, if n is a multi-valued natural number, then the exponent is taken in brackets. For example, 4^9 is another notation for the power of 4 9 . And here are some more examples of writing degrees using the symbol “^”: 14^(21) , (−2,1)^(155) . In what follows, we will primarily use degree notation of the form a n .

One of the problems inverse to raising to a power with a natural exponent is the problem of finding the base of a power from a known value of the power and a known exponent. This task leads to .

It is known that the set of rational numbers consists of integers and fractions, and each fraction can be represented as a positive or negative ordinary fraction. We defined a degree with an integer exponent in the previous paragraph, therefore, in order to complete the definition of a degree with a rational exponent, we need to give meaning to the degree of the number a with a fractional exponent m/n, where m is an integer and n is a natural number. Let's do it.

Let's consider a degree with a fractional exponent of the form . For the power-to-power property to remain valid, the equality must hold . If we take into account the resulting equality and how we determined , then it is logical to accept it provided that for given m, n and a the expression makes sense.

It is easy to check that for all properties of a degree with an integer exponent are valid (this was done in the section properties of a degree with a rational exponent).

The above reasoning allows us to make the following conclusion: if given m, n and a the expression makes sense, then the power of a with a fractional exponent m/n is called the nth root of a to the power of m.

This statement brings us close to the definition of a degree with a fractional exponent. All that remains is to describe at what m, n and a the expression makes sense. Depending on the restrictions placed on m, n and a, there are two main approaches.

The easiest way is to impose a constraint on a by taking a≥0 for positive m and a>0 for negative m (since for m≤0 the degree 0 of m is not defined). Then we get the following definition of a degree with a fractional exponent.

Definition.

Power of a positive number a with fractional exponent m/n, where m is an integer and n is a natural number, is called the nth root of the number a to the power m, that is, .

The fractional power of zero is also determined with the only caveat that the indicator must be positive.

Definition.

Power of zero with fractional positive exponent m/n, where m is a positive integer and n is a natural number, is defined as .
When the degree is not determined, that is, the degree of the number zero with a fractional negative exponent does not make sense.

It should be noted that with this definition of a degree with a fractional exponent, there is one caveat: for some negative a and some m and n, the expression makes sense, and we discarded these cases by introducing the condition a≥0. For example, the entries make sense or , and the definition given above forces us to say that powers with a fractional exponent of the form do not make sense, since the base should not be negative.

Another approach to determining a degree with a fractional exponent m/n is to separately consider even and odd exponents of the root. This approach requires an additional condition: the power of the number a, the exponent of which is , is considered to be the power of the number a, the exponent of which is the corresponding irreducible fraction (we will explain the importance of this condition below). That is, if m/n is an irreducible fraction, then for any natural number k the degree is first replaced by .

For even n and positive m, the expression makes sense for any non-negative a (an even root of a negative number does not make sense); for negative m, the number a must still be different from zero (otherwise there will be division by zero). And for odd n and positive m, the number a can be any (the root of an odd degree is defined for any real number), and for negative m, the number a must be different from zero (so that there is no division by zero).

The above reasoning leads us to this definition of a degree with a fractional exponent.

Definition.

Let m/n be an irreducible fraction, m an integer, and n a natural number. For any reducible fraction, the degree is replaced by . The power of a number with an irreducible fractional exponent m/n is for



Let us explain why a degree with a reducible fractional exponent is first replaced by a degree with an irreducible exponent. If we simply defined the degree as , and did not make a reservation about the irreducibility of the fraction m/n, then we would be faced with situations similar to the following: since 6/10 = 3/5, then the equality must hold , But , A .

Table of powers 2 (twos) from 0 to 32

The table below shows, in addition to powers of two, the maximum numbers that a computer can store for a given number of bits. Moreover, both for integers and signed numbers.

Historically, computers used the binary number system, and, accordingly, data storage. Thus, any number can be represented as a sequence of zeros and ones (bits of information). There are several ways to represent numbers as a binary sequence.

Let's consider the simplest of them - this is a positive integer. Then the larger the number we need to write, the longer the sequence of bits we need.

Below is table of powers of number 2. It will give us a representation of the required number of bits that we need to store numbers.

How to use table of powers of number two?

The first column is power of two, which simultaneously denotes the number of bits that represent the number.

Second column - value twos to the appropriate power (n).

An example of finding the power of 2. We find the number 7 in the first column. We look along the line to the right and find the value two to the seventh power(2 7) is 128

Third column - the maximum number that can be represented using a given number of bits(in the first column).

An example of determining the maximum unsigned integer. Using the data from the previous example, we know that 2 7 = 128. This is true if we want to understand what amount of numbers, can be represented using seven bits. But, since the first number is zero, then the maximum number that can be represented using seven bits is 128 - 1 = 127. This is the value of the third column.

Power of two (n)	Power of two value 2n	Maximum unsigned number written with n bits	Maximum signed number written with n bits
0	1	-	-
1	2	1	-
2	4	3	1
3	8	7	3
4	16	15	7
5	32	31	15
6	64	63	31
7	128	127	63
8	256	255	127
9	512	511	255
10	1 024	1 023	511
11	2 048	2 047	1023
12	40 96	4 095	2047
13	8 192	8 191	4095
14	16 384	16 383	8191
15	32 768	32 767	16383
16	65 536	65 535	32767
17	131 072	131 071	65 535
18	262 144	262 143	131 071
19	524 288	524 287	262 143
20	1 048 576	1 048 575	524 287
21	2 097 152	2 097 151	1 048 575
22	4 194 304	4 194 303	2 097 151
23	8 388 608	8 388 607	4 194 303
24	16 777 216	16 777 215	8 388 607
25	33 554 432	33 554 431	16 777 215
26	67 108 864	67 108 863	33 554 431
27	134 217 728	134 217 727	67 108 863
28	268 435 456	268 435 455	134 217 727
29	536 870 912	536 870 911	268 435 455
30	1 073 741 824	1 073 741 823	536 870 911
31	2 147 483 648	2 147 483 647	1 073 741 823
32	4 294 967 296	4 294 967 295	2 147 483 647

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.

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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 hundredths. 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 3 to the power - 2.

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 transform the value of the indicator into an ordinary fraction - 44, 89 2, 5 = 49, 89 5 2.

Now we perform 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 approximate value of 21, 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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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. Keep in mind that for negative numbers, raising to a non-integer power is undefined, so the calculator will report an error if you attempt it.

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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 .