Can the degree be negative? negative power

In one of the previous articles, we already mentioned the degree of a number. Today we will try to navigate in the process of finding its meaning. Scientifically speaking, we will figure out how to properly exponentiate. We will understand how this process is carried out, at the same time touching on all possible exponents: natural, irrational, rational, whole.

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

1. Concept definition.
2. Raising to negative art.
3. Whole score.
4. Raising a number to an irrational power.

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

Accordingly, the construction of 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 degree (0.6) 6 ″, then it can be simplified to the expression “Raise the number 0.6 to the power of 6”.

After that, you can proceed directly to the rules of construction.

Raising to a negative power

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

110 \u003d 0.1 \u003d 1 * 10 in minus 1 st.,

1100 \u003d 0.01 \u003d 1 * 10 in minus 2 steps.,

11000 \u003d 0.0001 \u003d 1 * 10 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 negative power. For this purpose, it is enough to simply shift the decimal component:

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

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

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

How to raise a number to a natural power

Recalling 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 * ... and divide by n times.

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

Raise -2 to the 4th tbsp.

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 it remains only to carry out the multiplication of integers: (-2) * (-2) * (-2) * (-2). We get 16.

Answer to the task:

(-2) in Art. 4=16.

Example:

Calculate the value: three point two sevenths squared.

This example is equal to the following product: three point two seventh times three point two seventh. Remembering how multiplication works mixed numbers, we complete the construction:

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

Answer: 10 39/49

With regard to the issue of raising to an irrational indicator, it should be noted that calculations begin to be carried out after the completion of the preliminary rounding of the basis of the degree to some rank, which would allow obtaining a 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 \u003d (3.14) 2 \u003d 9.8596. However, if we reduce P to ten-thousandths, we get P = 3.14159. Then squaring gets a completely different number: 9.8695877281.

It should be noted here that in many problems there is no need to raise irrational numbers to a power. As a rule, the answer is entered either in the form of, in fact, a degree, for example, the root of 6 to the power of 3, or, if the expression allows, its transformation is carried out: the root of 5 to 7 degrees \u003d 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 zero indicator;
• for a positive integer.

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

Now let's talk about the calculation of Art. 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 st. 0 will be equal to 1.

Accordingly, the construction of any real number to zero art. 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 the exponentiation 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. In the denominator of the fraction is Art. with a positive integer value, the value of which we have already learned to find. Now it remains only to consider an example of construction.

Example:

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

Solution Process:

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

The denominator is calculated simply: two cubed;

3 = 2*2*2=8.

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

One of the main characteristics in algebra, and indeed in all mathematics, is a degree. Of course, in the 21st century, all calculations can be carried out on an online calculator, but it is better to learn how to do it yourself for the development of brains.

In this article, we will consider the most important issues regarding this definition. Namely, we will understand what it is in general and what are its main functions, what properties exist in mathematics.

Let's look at examples of what the calculation looks like, what are the basic formulas. We will analyze the main types of quantities and how they differ from other functions.

Let's understand how to solve using this quantity various tasks. We will show with examples how to raise to a zero degree, irrational, negative, etc.

Online exponentiation calculator

What is the degree of a number

What is meant by the expression "raise a number to a power"?

The degree n of a number a is the product of factors of magnitude a n times in a row.

Mathematically it looks like this:

a n = a * a * a * …a n .

For example:

• 2 3 = 2 in the third step. = 2 * 2 * 2 = 8;
• 4 2 = 4 in step. two = 4 * 4 = 16;
• 5 4 = 5 in step. four = 5 * 5 * 5 * 5 = 625;
• 10 5 \u003d 10 in 5 step. = 10 * 10 * 10 * 10 * 10 = 100000;
• 10 4 \u003d 10 in 4 step. = 10 * 10 * 10 * 10 = 10000.

Below is a table of squares and cubes from 1 to 10.

Table of degrees from 1 to 10

Below are the results of raising natural numbers to positive powers - "from 1 to 100".

Degree properties

What is characteristic of such a mathematical function? Let's look at the basic properties.

Scientists have established the following signs characteristic of all degrees:

• a n * a m = (a) (n+m) ;
• a n: a m = (a) (n-m) ;
• (a b) m =(a) (b*m) .

Let's check with examples:

2 3 * 2 2 = 8 * 4 = 32. On the other hand 2 5 = 2 * 2 * 2 * 2 * 2 = 32.

Similarly: 2 3: 2 2 = 8 / 4 = 2. Otherwise 2 3-2 = 2 1 =2.

(2 3) 2 = 8 2 = 64. What if it's different? 2 6 = 2 * 2 * 2 * 2 * 2 * 2 = 32 * 2 = 64.

As you can see, the rules work.

But how to be with addition and subtraction? Everything is simple. First exponentiation is performed, and only then addition and subtraction.

Let's look at examples:

• 3 3 + 2 4 = 27 + 16 = 43;
• 5 2 - 3 2 = 25 - 9 = 16

But in this case, you must first calculate the addition, since there are actions in brackets: (5 + 3) 3 = 8 3 = 512.

How to produce computing in more difficult cases ? The order is the same:

• if there are brackets, you need to start with them;
• then exponentiation;
• then perform operations of multiplication, division;
• after addition, subtraction.

There are specific properties that are not characteristic of all degrees:

1. The root of the nth degree from the number a to the degree m will be written as: a m / n .
2. When raising a fraction to a power: both the numerator and its denominator are subject to this procedure.
3. When raising the product of different numbers to a power, the expression will correspond to the product of these numbers to a given power. That is: (a * b) n = a n * b n .
4. When raising a number to a negative power, you need to divide 1 by a number in the same step, but with a “+” sign.
5. If the denominator of a fraction is in a negative power, then this expression will be equal to the product of the numerator and the denominator in a positive power.
6. Any number to the power of 0 = 1, and to the step. 1 = to himself.

These rules are important in individual cases, we will consider them in more detail below.

Degree with a negative exponent

What to do with a negative degree, that is, when the indicator is negative?

Based on properties 4 and 5(see point above) it turns out:

A (- n) \u003d 1 / A n, 5 (-2) \u003d 1/5 2 \u003d 1/25.

And vice versa:

1 / A (- n) \u003d A n, 1 / 2 (-3) \u003d 2 3 \u003d 8.

What if it's a fraction?

(A / B) (- n) = (B / A) n , (3 / 5) (-2) = (5 / 3) 2 = 25 / 9.

Degree with a natural indicator

It is understood as a degree with exponents equal to integers.

Things to remember:

A 0 = 1, 1 0 = 1; 2 0 = 1; 3.15 0 = 1; (-4) 0 = 1…etc.

A 1 = A, 1 1 = 1; 2 1 = 2; 3 1 = 3…etc.

Also, if (-a) 2 n +2 , n=0, 1, 2…then the result will be with a “+” sign. If a negative number is raised to an odd power, then vice versa.

General properties, and all the specific features described above, are also characteristic of them.

Fractional degree

This view can be written as a scheme: A m / n. It is read as: the root of the nth degree of the number A to the power of m.

With a fractional indicator, you can do anything: reduce, decompose into parts, raise to another degree, etc.

Degree with irrational exponent

Let α be an irrational number and А ˃ 0.

To understand the essence of the degree with such an indicator, Let's look at different possible cases:

• A \u003d 1. The result will be equal to 1. Since there is an axiom - 1 is equal to one in all powers;

А r 1 ˂ А α ˂ А r 2 , r 1 ˂ r 2 are rational numbers;

• 0˂А˂1.

In this case, vice versa: А r 2 ˂ А α ˂ А r 1 under the same conditions as in the second paragraph.

For example, the exponent is the number π. It is rational.

r 1 - in this case it is equal to 3;

r 2 - will be equal to 4.

Then, for A = 1, 1 π = 1.

A = 2, then 2 3 ˂ 2 π ˂ 2 4 , 8 ˂ 2 π ˂ 16.

A = 1/2, then (½) 4 ˂ (½) π ˂ (½) 3 , 1/16 ˂ (½) π ˂ 1/8.

Such degrees are characterized by all the mathematical operations and specific properties described above.

Conclusion

Let's summarize - what are these values ​​for, what are the advantages of such functions? Of course, first of all, they simplify the lives of mathematicians and programmers when solving examples, since they allow minimizing calculations, reducing algorithms, systematizing data, and much more.

Where else can this knowledge be useful? At any working specialty: medicine, pharmacology, dentistry, construction, technology, engineering, design, etc.

Raising to a negative power is one of the basic elements of mathematics, which is often encountered in solving algebraic problems. Below is a detailed instruction.

How to raise to a negative power - theory

When we take a number to the usual power, we multiply its value several times. For example, 3 3 \u003d 3 × 3 × 3 \u003d 27. With a negative fraction, the opposite is true. General form according to the formula will have the following form: a -n = 1/a n . Thus, to raise a number to a negative power, you need to divide one by the given number, but already to a positive power.

How to raise to a negative power - examples on ordinary numbers

With the above rule in mind, let's solve a few examples.

4 -2 = 1/4 2 = 1/16
Answer: 4 -2 = 1/16

4 -2 = 1/-4 2 = 1/16.
The answer is -4 -2 = 1/16.

But why is the answer in the first and second examples the same? The fact is that when a negative number is raised to an even power (2, 4, 6, etc.), the sign becomes positive. If the degree were even, then the minus is preserved:

4 -3 = 1/(-4) 3 = 1/(-64)

How to raise to a negative power - numbers from 0 to 1

Recall that when a number between 0 and 1 is raised to a positive power, the value decreases as the power increases. So for example, 0.5 2 = 0.25. 0.25

Example 3: Calculate 0.5 -2
Solution: 0.5 -2 = 1/1/2 -2 = 1/1/4 = 1×4/1 = 4.
Answer: 0.5 -2 = 4

Parsing (sequence of actions):

• Translation decimal 0.5 to fractional 1/2. It's easier.
  Raise 1/2 to a negative power. 1/(2) -2 . Divide 1 by 1/(2) 2 , we get 1/(1/2) 2 => 1/1/4 = 4

Example 4: Calculate 0.5 -3
Solution: 0.5 -3 = (1/2) -3 = 1/(1/2) 3 = 1/(1/8) = 8

Example 5: Calculate -0.5 -3
Solution: -0.5 -3 = (-1/2) -3 = 1/(-1/2) 3 = 1/(-1/8) = -8
Answer: -0.5 -3 = -8

Based on the 4th and 5th examples, we will draw several conclusions:

• For a positive number between 0 and 1 (example 4) raised to a negative power, whether the power is even or odd, the value of the expression will be positive. In this case, the greater the degree, the greater the value.
• For a negative number between 0 and 1 (Example 5), raised to a negative power, whether the power is even or odd, the value of the expression will be negative. In this case, the higher the degree, the lower the value.

How to raise to a negative power - the power as a fractional number

Expressions of this type have the following form: a -m/n , where a is an ordinary number, m is the numerator of the degree, n is the denominator of the degree.

Consider an example:
Calculate: 8 -1/3

Solution (sequence of actions):

• Remember the rule for raising a number to a negative power. We get: 8 -1/3 = 1/(8) 1/3 .
• Note that the denominator is 8 to a fractional power. The general form of calculating a fractional degree is as follows: a m/n = n √8 m .
• Thus, 1/(8) 1/3 = 1/(3 √8 1). We get the cube root of eight, which is 2. Based on this, 1/(8) 1/3 = 1/(1/2) = 2.
• Answer: 8 -1/3 = 2

From school, we all know the rule about raising to a power: any number with an exponent N is equal to the result of multiplying this number by itself N times. In other words, 7 to the power of 3 is 7 multiplied by itself three times, that is, 343. Another rule - raising any value to the power of 0 gives one, and raising a negative value is the result of ordinary exponentiation, if it is even, and the same result with a minus sign if it is odd.

The rules also give an answer on how to raise a number to a negative power. To do this, you need to raise the required value by the module of the indicator in the usual way, and then divide the unit by the result.

From these rules, it becomes clear that the implementation of real problems with large quantities will require the presence of technical means. Manually it will be possible to multiply by itself a maximum range of numbers up to twenty or thirty, and then no more than three or four times. This is not to mention the fact that then also divide the unit by the result. Therefore, for those who do not have a special engineering calculator at hand, we will tell you how to raise a number to a negative power in Excel.

Solving problems in Excel

To solve problems with exponentiation, Excel allows you to use one of two options.

The first is the use of the formula with the standard cap symbol. Enter the following data in the worksheet cells:

In the same way, you can raise the desired value to any power - negative, fractional. Let's do the following and answer the question of how to raise a number to a negative power. Example:

It is possible to correct directly in the formula =B2^-C2.

The second option is to use the ready-made "Degree" function, which takes two mandatory arguments - a number and an indicator. To start using it, it is enough to put an equal sign (=) in any free cell, indicating the beginning of the formula, and enter the above words. It remains to select two cells that will participate in the operation (or specify specific numbers manually), and press the Enter key. Let's look at a few simple examples.

As you can see, there is nothing complicated about how to raise a number to a negative power and to a regular one using Excel. After all, to solve this problem, you can use both the familiar “lid” symbol and the easy-to-remember built-in function of the program. This is a definite plus!

Let's move on to more complex examples. Let's recall the rule on how to raise a number to a negative power of a fractional character, and we will see that this task is very simply solved in Excel.

Fractional indicators

In short, the algorithm for calculating a number with a fractional exponent is as follows.

1. Convert a fractional exponent to a proper or improper fraction.
2. Raise our number to the numerator of the resulting converted fraction.
3. From the number obtained in the previous paragraph, calculate the root, with the condition that the root indicator will be the denominator of the fraction obtained in the first stage.

Agree that even when operating with small numbers and proper fractions, such calculations can take a lot of time. It's good that the spreadsheet processor Excel does not care what number and to what degree to raise. Try solving the following example in an Excel worksheet:

Using the above rules, you can check and make sure that the calculation is correct.

At the end of our article, we will give in the form of a table with formulas and results several examples of how to raise a number to a negative power, as well as several examples with fractional numbers and powers.

Example table

Check the Excel worksheet for the following examples. For everything to work correctly, you need to use a mixed reference when copying the formula. Fix the number of the column containing the number being raised, and the number of the row containing the indicator. Your formula should look something like this: "=$B4^C$3".

Please note that positive numbers (even non-integer ones) are calculated without problems for any exponents. There are no problems with raising any numbers to integers. But raising a negative number to a fractional power will turn out to be a mistake for you, since it is impossible to follow the rule indicated at the beginning of our article about raising negative numbers, because evenness is a characteristic of an exclusively INTEGER number.

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

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

a-n = (1 / a n)

The properties of negative values ​​of powers of numbers are similar to powers with a positive exponent. Represented Equation a m / a n = a m-n can be fair as

« Nowhere, as in mathematics, the clarity and accuracy of the conclusion does not allow a person to get away from the answer by talking around the question.».

A. D. Alexandrov

at n more m , as well as 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 indicator of the degree b - desired numerical value, a - the base of the degree as a natural numerical value. Then determine the module, that is, the absolute value of a negative number, which acts as an exponent. Calculate the degree of the given number relative to the absolute number as an indicator. The value of the degree is found by dividing one by the resulting number.

Rice. one

Consider the power of a number with a negative fractional exponent. Imagine that the number a is any positive number, the numbers n and m - integers. By definition a , which is raised to the power - equals one divided by the same number with a positive degree (Fig. 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 began 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 ​​of numbers were systematically used when solving problems 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 directions of segments. It was Descartes who suggested that the number raised to a power be displayed as a two-story formula a n .

Power 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. Multiplying powers with the same base their scores are:

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 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 a to a degree m/n, you need to extract the root n th degree of m th power of this number a.

Exponentiation is an operation closely related to multiplication, this operation is the result of multiple multiplication of a number by itself. Let's represent the formula: a1 * a2 * ... * an = an.

For example, a=2, n=3: 2 * 2 * 2=2^3 = 8 .

In general, exponentiation is often used in various formulas in mathematics and physics. This function has a more scientific purpose than the four basic ones: Addition, Subtraction, Multiplication, Division.

Raising a number to a power

Raising a number to a power is not a difficult operation. It is related to multiplication like the relationship between multiplication and addition. Record an - a short record of the n-th number of numbers "a" multiplied by each other.

Consider exponentiation on the simplest examples, moving on to complex ones.

For example, 42. 42 = 4 * 4 = 16 . Four squared (to the second power) equals sixteen. If you do not understand the multiplication 4 * 4, then read our article about multiplication.

Let's look at another example: 5^3. 5^3 = 5 * 5 * 5 = 25 * 5 = 125 . Five cubed (to the third power) equals one hundred and twenty-five.

Another example: 9^3. 9^3 = 9 * 9 * 9 = 81 * 9 = 729 . Nine cubed equals seven hundred twenty-nine.

Exponentiation Formulas

To correctly raise to a power, you need to remember and know the formulas below. There is nothing beyond natural in this, the main thing is to understand the essence and then they will not only be remembered, but also seem easy.

Raising a monomial to a power

What is a monomial? This is the product of numbers and variables in any quantity. For example, two is a monomial. And this article is about raising such monomials to a power.

Using exponentiation formulas, it will not be difficult to calculate the exponentiation of a monomial to a power.

For example, (3x^2y^3)^2= 3^2 * x^2 * 2 * y^(3 * 2) = 9x^4y^6; If you raise a monomial to a power, then each component of the monomial is raised to a power.

When raising a variable that already has a degree to a power, the degrees are multiplied. For example, (x^2)^3 = x^(2 * 3) = x^6 ;

Raising to a negative power

negative power is the reverse number. What is a reciprocal? For any number X, the reciprocal is 1/X. That is X-1=1/X. This is the essence of the negative degree.

Consider the example (3Y)^-3:

(3Y)^-3 = 1/(27Y^3).

Why is that? Since there is a minus in the degree, we simply transfer this expression to the denominator, and then raise it to the third power. Just right?

Raising to a fractional power

Let's start the discussion on specific example. 43/2. What does power 3/2 mean? 3 - numerator, means raising a number (in this case 4) in cube. The number 2 is the denominator, this is the extraction of the second root of the number (in this case 4).

Then we get the square root of 43 = 2^3 = 8 . Answer: 8.

So, the denominator of a fractional degree can be either 3 or 4, and to infinity any number, and this number determines the degree square root extracted from the given number. Of course, the denominator cannot be zero.

Raising a root to a power

If the root is raised to a power equal to the power of the root itself, then the answer is the radical expression. For example, (√x)2 = x. And so in any case of equality of the degree of the root and the degree of raising the root.

If (√x)^4. Then (√x)^4=x^2. To check the solution, we translate the expression into an expression with a fractional degree. Since the root is square, the denominator is 2. And if the root is raised to the fourth power, then the numerator is 4. We get 4/2=2. Answer: x = 2.

Anyway the best option just convert the expression to an expression with a fractional power. If the fraction is not reduced, then such an answer will be, provided that the root of the given number is not allocated.

Exponentiation of a complex number

What is a complex number? A complex number is an expression that has the formula a + b * i; a, b are real numbers. i is the number that, when squared, gives the number -1.

Consider an example. (2 + 3i)^2.

(2 + 3i)^2 = 22 +2 * 2 * 3i +(3i)^2 = 4+12i^-9=-5+12i.

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Exponentiation online

With the help of our calculator, you can calculate the exponentiation of a number to a power:

Exponentiation Grade 7

Raising to a power begins to pass schoolchildren only in the seventh grade.

Exponentiation is an operation closely related to multiplication, this operation is the result of multiple multiplication of a number by itself. Let's represent the formula: a1 * a2 * … * an=an .

For example, a=2, n=3: 2 * 2 * 2 = 2^3 = 8.

Solution Examples:

Exponentiation presentation

Presentation on exponentiation, designed for seventh graders. The presentation may clarify some incomprehensible points, but there will probably not be such points thanks to our article.

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