What does a number with a negative power mean? How to raise a number to a negative power - examples with descriptions in Excel

In one of the previous articles we already mentioned the power of a number. Today we will try to navigate the process of finding its meaning. Scientifically speaking, we will figure out how to raise to a power correctly. We will figure out how this process is carried out, and at the same time we will touch on all possible exponents: natural, irrational, rational, integer.

So, let's take a closer look at the solutions to the examples and find out what it means:

Definition of the concept.
Raising to negative art.
A whole indicator.
Raising a number to an irrational power.

Here is a definition that accurately reflects the meaning: “Exponentiation is the definition of the value of a power of a number.”

Accordingly, raising the number a in Art. r and the process of finding the value of the degree a with the exponent r are identical concepts. For example, if the task is to calculate the value of the power (0.6)6″, then it can be simplified to the expression “Raise the number 0.6 to the power of 6.”

After this, you can proceed directly to the construction rules.

Raising to a negative power

For clarity, you should pay attention to the following chain of expressions:

110=0.1=1* 10 minus 1 tbsp.,

1100=0.01=1*10 in minus 2 degrees,

11000=0.0001=1*10 in minus 3 st.,

110000=0.00001=1*10 to minus 4 degrees.

Thanks to these examples, you can clearly see the ability to instantly calculate 10 to any minus power. For this purpose, it is enough to simply shift the decimal component:

10 to the -1 degree - before one there is 1 zero;
in -3 - three zeros before one;
in -9 there are 9 zeros and so on.

It is also easy to understand from this diagram how much 10 minus 5 tbsp will be. -

1100000=0,000001=(1*10)-5.

How to raise a number to a natural power

Remembering the definition, we take into account that the natural number a in Art. n equals the product of n factors, each of which equals a. Let's illustrate: (a*a*…a)n, where n is the number of numbers that are multiplied. Accordingly, in order to raise a to n, it is necessary to calculate the product of the following form: a*a*…a divided by n times.

From this it becomes obvious that raising to natural st. relies on the ability to perform multiplication(this material is covered in the section on multiplying real numbers). Let's look at the problem:

Raise -2 to the 4th st.

We are dealing with a natural indicator. Accordingly, the course of the decision will be as follows: (-2) in Art. 4 = (-2)*(-2)*(-2)*(-2). Now all that remains is to multiply the integers: (-2)*(-2)*(-2)*(-2). We get 16.

Answer to the problem:

(-2) in Art. 4=16.

Example:

Calculate the value: three point two sevenths squared.

This example equals the following product: three point two sevenths multiplied by three point two sevenths. Recalling how mixed numbers are multiplied, we complete the construction:

3 point 2 sevenths multiplied by themselves;
equals 23 sevenths multiplied by 23 sevenths;
equals 529 forty-ninths;
we reduce and we get 10 thirty-nine forty-ninths.

Answer: 10 39/49

Regarding the issue of raising to an irrational exponent, it should be noted that calculations begin to be carried out after the completion of preliminary rounding of the basis of the degree to any digit that would allow obtaining the value with a given accuracy. For example, we need to square the number P (pi).

We start by rounding P to hundredths and get:

P squared = (3.14)2 = 9.8596. However, if we reduce P to ten thousandths, we get P = 3.14159. Then squaring gives a completely different number: 9.8695877281.

It should be noted here that in many problems there is no need to raise irrational numbers to powers. As a rule, the answer is entered either in the form of the actual degree, for example, the root of 6 to the power of 3, or, if the expression allows, its transformation is carried out: root of 5 to 7 degrees = 125 root of 5.

How to raise a number to an integer power

This algebraic manipulation is appropriate take into account for the following cases:

for integers;
for a zero indicator;
for a positive integer exponent.

Since almost all positive integers coincide with the mass of natural numbers, setting to a positive integer power is the same process as setting in Art. natural. We described this process in the previous paragraph.

Now let's talk about calculating st. null. We have already found out above that the zero power of the number a can be determined for any non-zero a (real), while a in Art. 0 will equal 1.

Accordingly, raising any real number to the zero st. will give one.

For example, 10 in st. 0=1, (-3.65)0=1, and 0 in st. 0 cannot be determined.

In order to complete raising to an integer power, it remains to decide on the options for negative integer values. We remember that Art. from a with an integer exponent -z will be defined as a fraction. The denominator of the fraction is st. with a positive integer value, the value of which we have already learned to find. Now all that remains is to consider an example of construction.

Example:

Calculate the value of the number 2 cubed with a negative integer exponent.

Solution process:

According to the definition of a degree with a negative exponent, we denote: two minus 3 degrees. equals one to two to the third power.

The denominator is calculated simply: two cubed;

3 = 2*2*2=8.

Answer: two to the minus 3rd st. = one eighth.

A number raised to a power They call a number that is multiplied by itself several times.

Power of a number with a negative value (a - n) can be determined in a similar way to how the power of the same number with a positive exponent is determined (a n) . However, it also requires additional definition. The formula is defined as:

a-n = (1/a n)

The properties of negative powers of numbers are similar to powers with a positive exponent. Presented equation a m/a n= a m-n may be fair as

« Nowhere, as in mathematics, does the clarity and accuracy of the conclusion allow a person to wriggle out of an answer by talking around the question».

A. D. Alexandrov

at n more m , and with m more n . Let's look at an example: 7 2 -7 5 =7 2-5 =7 -3 .

First you need to determine the number that acts as a definition of the degree. b=a(-n) . In this example -n is an exponent b - the desired numerical value, a - the base of the degree in the form of a natural numeric value. Then determine the module, that is, the absolute value of a negative number, which acts as an exponent. Calculate the degree of a given number relative to an absolute number, as an indicator. The value of the degree is found by dividing one by the resulting number.

Rice. 1

Consider the power of a number with a negative fractional exponent. Let's imagine that the number a is any positive number, numbers n And m - integers. According to definition a , which is raised to the power - equals one divided by the same number with a positive power (Figure 1). When the power of a number is a fraction, then in such cases only numbers with positive exponents are used.

Worth remembering that zero can never be an exponent of a number (the rule of division by zero).

The spread of such a concept as a number became such manipulations as measurement calculations, as well as the development of mathematics as a science. The introduction of negative values was due to the development of algebra, which gave general solutions arithmetic problems, regardless of their specific meaning and initial numerical data. In India back in the 6th-11th centuries negative values numbers were systematically used during problem solving and were interpreted in the same way as today. In European science, negative numbers began to be widely used thanks to R. Descartes, who gave a geometric interpretation of negative numbers as the directions of segments. It was Descartes who proposed the designation of a number raised to a power to be displayed as a two-story formula a n .

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. It should be remembered that for negative numbers Raising to a non-integer power is undefined and the calculator will therefore report an error if you attempt it.

Degree calculator

Raise to power

Exponentiations: 20880

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 .

Continuing the conversation about the power of a number, it is logical to figure out how to find the value of the power. This process is called exponentiation. In this article we will study how exponentiation is performed, while we will touch on all possible exponents - natural, integer, rational and irrational. And according to tradition, we will consider in detail solutions to examples of raising numbers to various powers.

Page navigation.

What does "exponentiation" mean?

Let's start by explaining what is called exponentiation. Here is the relevant definition.

Definition.

Exponentiation- this is finding the value of the power of a number.

Thus, finding the value of the power of a number a with exponent r and raising the number a to the power r are the same thing. For example, if the task is “calculate the value of the power (0.5) 5,” then it can be reformulated as follows: “Raise the number 0.5 to the power 5.”

Now you can go directly to the rules by which exponentiation is performed.

Raising a number to a natural power

In practice, equality based on is usually applied in the form . That is, when raising a number a to a fractional power m/n, first the nth root of the number a is taken, after which the resulting result is raised to an integer power m.

Let's look at solutions to examples of raising to a fractional power.

Example.

Calculate the value of the degree.

Solution.

We will show two solutions.

First way. By definition of a degree with a fractional exponent. We calculate the value of the degree under the root sign, and then extract the cube root: .

Second way. By the definition of a degree with a fractional exponent and based on the properties of the roots, the following equalities are true: . Now we extract the root , finally, we raise it to an integer power .

Obviously, the obtained results of raising to a fractional power coincide.

Answer:

Note that the fractional exponent can be written as decimal or a mixed number, in these cases it should be replaced with the corresponding ordinary fraction, and then raised to a power.

Example.

Calculate (44.89) 2.5.

Solution.

Let us write the exponent in the form common fraction(if necessary, see the article): . Now we perform the raising to a fractional power:

Answer:

(44,89) 2,5 =13 501,25107 .

It should also be said that raising numbers to rational powers is a rather labor-intensive process (especially when the numerator and denominator of the fractional exponent contain sufficiently large numbers), which is usually carried out using computer technology.

To conclude this point, we will focus on raising the number zero to a fractional power. We gave the following meaning to the fractional power of zero of the form: when we have , and at zero to the power m/n is not defined. So, zero to a fractional positive power is zero, for example, . And zero in a fractional negative power does not make sense, for example, the expressions 0 -4.3 do not make sense.

Raising to an irrational power

Sometimes it becomes necessary to find out the value of the power of a number with an irrational exponent. In this case, for practical purposes it is usually sufficient to obtain the value of the degree accurate to a certain sign. Let us immediately note that in practice this value is calculated using electronic computers, since raising it to an irrational power manually requires large quantity cumbersome calculations. But we will still describe in general terms the essence of the actions.

To obtain an approximate value of the power of a number a with an irrational exponent, some decimal approximation of the exponent is taken and the value of the power is calculated. This value is an approximate value of the power of the number a with an irrational exponent. The more accurate the decimal approximation of a number is taken initially, the more exact value degree will be obtained in the end.

As an example, let's calculate the approximate value of the power of 2 1.174367... . Let's take the following decimal approximation of the irrational exponent: . Now we raise 2 to the rational power 1.17 (we described the essence of this process in the previous paragraph), we get 2 1.17 ≈2.250116. Thus, 2 1,174367... ≈2 1,17 ≈2,250116 . If we take a more accurate decimal approximation of the irrational exponent, for example, then we obtain a more accurate value of the original exponent: 2 1,174367... ≈2 1,1743 ≈2,256833 .

Bibliography.

Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics textbook for 5th grade. educational institutions.
Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 7th grade. educational institutions.
Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8th grade. educational institutions.
Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 9th grade. educational institutions.
Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the beginnings of analysis: Textbook for grades 10 - 11 of general education institutions.
Gusev V.A., Mordkovich A.G. Mathematics (a manual for those entering technical schools).

The power is used to simplify the operation of multiplying a number by itself. For example, instead of writing, you can write 4 5 (\displaystyle 4^(5))(an explanation for this transition is given in the first section of this article). Degrees make it easier to write long or complex expressions or equations; powers are also easy to add and subtract, resulting in a simplified expression or equation (for example, 4 2 ∗ 4 3 = 4 5 (\displaystyle 4^(2)*4^(3)=4^(5))).

Note: if you need to solve an exponential equation (in such an equation the unknown is in the exponent), read.

Steps

Solving simple problems with degrees

Multiply the base of the exponent by itself a number of times equal to the exponent. If you need to solve a power problem by hand, rewrite the power as a multiplication operation, where the base of the power is multiplied by itself. For example, given a degree 3 4 (\displaystyle 3^(4)). In this case, the base of power 3 must be multiplied by itself 4 times: 3 ∗ 3 ∗ 3 ∗ 3 (\displaystyle 3*3*3*3). Here are other examples:

First, multiply the first two numbers. For example, 4 5 (\displaystyle 4^(5)) = 4 ∗ 4 ∗ 4 ∗ 4 ∗ 4 (\displaystyle 4*4*4*4*4). Don't worry - the calculation process is not as complicated as it seems at first glance. First multiply the first two fours and then replace them with the result. Like this:

4 5 = 4 ∗ 4 ∗ 4 ∗ 4 ∗ 4 (\displaystyle 4^(5)=4*4*4*4*4)
- 4 ∗ 4 = 16 (\displaystyle 4*4=16)

Multiply the result (16 in our example) by the next number. Each subsequent result will increase proportionally. In our example, multiply 16 by 4. Like this:
- 4 5 = 16 ∗ 4 ∗ 4 ∗ 4 (\displaystyle 4^(5)=16*4*4*4)
  - 16 ∗ 4 = 64 (\displaystyle 16*4=64)
- 4 5 = 64 ∗ 4 ∗ 4 (\displaystyle 4^(5)=64*4*4)
  - 64 ∗ 4 = 256 (\displaystyle 64*4=256)
- 4 5 = 256 ∗ 4 (\displaystyle 4^(5)=256*4)
  - 256 ∗ 4 = 1024 (\displaystyle 256*4=1024)
- Continue multiplying the result of the first two numbers by the next number until you get your final answer. To do this, multiply the first two numbers, and then multiply the resulting result by the next number in the sequence. This method is valid for any degree. In our example you should get: 4 5 = 4 ∗ 4 ∗ 4 ∗ 4 ∗ 4 = 1024 (\displaystyle 4^(5)=4*4*4*4*4=1024) .
Solve the following problems. Check your answer using a calculator.
- 8 2 (\displaystyle 8^(2))
- 3 4 (\displaystyle 3^(4))
- 10 7 (\displaystyle 10^(7))
On your calculator, look for the key labeled "exp" or " x n (\displaystyle x^(n))", or "^". Using this key you will raise a number to a power. It is almost impossible to calculate a degree with a large indicator manually (for example, the degree 9 15 (\displaystyle 9^(15))), but the calculator can easily cope with this task. In Windows 7, the standard calculator can be switched to engineering mode; To do this, click “View” -> “Engineering”. To switch to normal mode, click “View” -> “Normal”.
- Check your answer using search engine(Google or Yandex). Using the "^" key on your computer keyboard, enter the expression into the search engine, which will instantly display the correct answer (and possibly suggest similar expressions for you to study).
Addition, subtraction, multiplication of powers
1. You can add and subtract degrees only if they have the same bases. If you need to add powers with the same bases and exponents, then you can replace the addition operation with the multiplication operation. For example, given the expression 4 5 + 4 5 (\displaystyle 4^(5)+4^(5)). Remember that the degree 4 5 (\displaystyle 4^(5)) can be represented in the form 1 ∗ 4 5 (\displaystyle 1*4^(5)); Thus, 4 5 + 4 5 = 1 ∗ 4 5 + 1 ∗ 4 5 = 2 ∗ 4 5 (\displaystyle 4^(5)+4^(5)=1*4^(5)+1*4^(5) =2*4^(5))(where 1 +1 =2). That is, count the number of similar degrees, and then multiply that degree and this number. In our example, raise 4 to the fifth power, and then multiply the resulting result by 2. Remember that the addition operation can be replaced by the multiplication operation, for example, 3 + 3 = 2 ∗ 3 (\displaystyle 3+3=2*3). Here are other examples:
  - 3 2 + 3 2 = 2 ∗ 3 2 (\displaystyle 3^(2)+3^(2)=2*3^(2))
  - 4 5 + 4 5 + 4 5 = 3 ∗ 4 5 (\displaystyle 4^(5)+4^(5)+4^(5)=3*4^(5))
  - 4 5 − 4 5 + 2 = 2 (\displaystyle 4^(5)-4^(5)+2=2)
  - 4 x 2 − 2 x 2 = 2 x 2 (\displaystyle 4x^(2)-2x^(2)=2x^(2))
2. When multiplying powers with the same base, their exponents are added (the base does not change). For example, given the expression x 2 ∗ x 5 (\displaystyle x^(2)*x^(5)). In this case, you just need to add the indicators, leaving the base unchanged. Thus, x 2 ∗ x 5 = x 7 (\displaystyle x^(2)*x^(5)=x^(7)). Here is a visual explanation of this rule:
  
  When raising a power to a power, the exponents are multiplied. For example, a degree is given. Since exponents are multiplied, then (x 2) 5 = x 2 ∗ 5 = x 10 (\displaystyle (x^(2))^(5)=x^(2*5)=x^(10)). The point of this rule is that you multiply by powers (x 2) (\displaystyle (x^(2))) on itself five times. Like this:
  - (x 2) 5 (\displaystyle (x^(2))^(5))
  - (x 2) 5 = x 2 ∗ x 2 ∗ x 2 ∗ x 2 ∗ x 2 (\displaystyle (x^(2))^(5)=x^(2)*x^(2)*x^( 2)*x^(2)*x^(2))
  - Since the base is the same, the exponents simply add up: (x 2) 5 = x 2 ∗ x 2 ∗ x 2 ∗ x 2 ∗ x 2 = x 10 (\displaystyle (x^(2))^(5)=x^(2)*x^(2)* x^(2)*x^(2)*x^(2)=x^(10))
3. A power with a negative exponent should be converted to a fraction (reverse power). It doesn't matter if you don't know what a reciprocal degree is. If you are given a degree with a negative exponent, e.g. 3 − 2 (\displaystyle 3^(-2)), write this degree in the denominator of the fraction (put 1 in the numerator), and make the exponent positive. In our example: 1 3 2 (\displaystyle (\frac (1)(3^(2)))). Here are other examples:
  
  When dividing degrees with the same base, their exponents are subtracted (the base does not change). The division operation is the opposite of the multiplication operation. For example, given the expression 4 4 4 2 (\displaystyle (\frac (4^(4))(4^(2)))). Subtract the exponent in the denominator from the exponent in the numerator (do not change the base). Thus, 4 4 4 2 = 4 4 − 2 = 4 2 (\displaystyle (\frac (4^(4))(4^(2)))=4^(4-2)=4^(2)) = 16 .
  - The power in the denominator can be written as follows: 1 4 2 (\displaystyle (\frac (1)(4^(2)))) = 4 − 2 (\displaystyle 4^(-2)). Remember that a fraction is a number (power, expression) with a negative exponent.
4. Below are some expressions that will help you learn to solve problems with exponents. The expressions given cover the material presented in this section. To see the answer, simply select the empty space after the equals sign.
Solving problems with fractional exponents
1. If the exponent is an improper fraction, then the exponent can be decomposed into two powers to simplify the solution of the problem. There is nothing complicated about this - just remember the rule of multiplying powers. For example, a degree is given. Convert such a power into a root whose power is equal to the denominator of the fractional exponent, and then raise this root to a power equal to the numerator of the fractional exponent. To do this, remember that = 5 3 (\displaystyle (\frac (5)(3)))(1 3) ∗ 5 (\displaystyle ((\frac (1)(3)))*5)
  - . In our example:
  - x 5 3 (\displaystyle x^(\frac (5)(3)))
  - x 1 3 = x 3 (\displaystyle x^(\frac (1)(3))=(\sqrt[(3)](x))) = x 5 3 = x 5 ∗ x 1 3 (\displaystyle x^(\frac (5)(3))=x^(5)*x^(\frac (1)(3)))
2. (x 3) 5 (\displaystyle ((\sqrt[(3)](x)))^(5))
3. Some calculators have a button to calculate exponents (you must first enter the base, then press the button, and then enter the exponent). It is denoted as ^ or x^y. Remember that any number to the first power is equal to itself, for example, 4 1 = 4. (\displaystyle 4^(1)=4.) Moreover, any number multiplied or divided by one is equal to itself, e.g. 5 ∗ 1 = 5 (\displaystyle 5*1=5) And.
4. 5 / 1 = 5 (\displaystyle 5/1=5) Know that the power 0 0 does not exist (such a power has no solution). If you try to solve such a degree on a calculator or on a computer, you will receive an error. But remember that any number to the zero power is 1, for example,
5. 4 0 = 1. (\displaystyle 4^(0)=1.) In higher mathematics, which operates with imaginary numbers: e a i x = c o s a x + i s i n a x (\displaystyle e^(a)ix=cosax+isinax) , Where i = (− 1) (\displaystyle i=(\sqrt (())-1))
- Warnings