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

In this article we will look at the most important issues related to this definition. Namely, we will understand what it is in general and what its main functions are, what properties there are in mathematics.

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

Let us understand how to solve various problems using this quantity. We will show with examples how to raise to the zero power, irrational, negative, etc.

Online exponentiation calculator

What is a power of a number

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

The power n of a number 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 degree. = 2 * 2 * 2 = 8;
• 4 2 = 4 to step. two = 4 * 4 = 16;
• 5 4 = 5 to step. four = 5 * 5 * 5 * 5 = 625;
• 10 5 = 10 in 5 steps. = 10 * 10 * 10 * 10 * 10 = 100000;
• 10 4 = 10 in 4 steps. = 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 construction natural numbers to positive powers – “from 1 to 100”.

Ch-lo	2nd st.	3rd stage
1	1	1
2	4	8
3	9	27
4	16	64
5	25	125
6	36	216
7	49	343
8	64	512
9	81	279
10	100	1000

Properties of degrees

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 what about with addition and subtraction? It's simple. Exponentiation is performed first, and then addition and subtraction.

Let's look at examples:

• 3 3 + 2 4 = 27 + 16 = 43;
• 5 2 – 3 2 = 25 – 9 = 16. Please note: the rule will not hold if you subtract first: (5 – 3) 2 = 2 2 = 4.

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

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

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

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

1. The nth root of a number a to the m degree 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 constructing a work different numbers to a power, the expression will correspond to the product of these numbers to the 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 century, but with a “+” sign.
5. If the denominator of a fraction is to a negative power, then this expression will be equal to the product of the numerator and the denominator to a positive power.
6. Any number to the power of 0 = 1, and to the power of. 1 = to yourself.

These rules are important in some cases; we will consider them in more detail below.

Degree with a negative exponent

What to do with a minus degree, i.e. when the indicator is negative?

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

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

And vice versa:

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

What if it's a fraction?

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

Degree with 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.

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

General properties and that's it specific signs, described above, are also characteristic of them.

Fractional degree

This type can be written as a scheme: A m / n. Read as: the nth root of the number A to the power m.

WITH fractional indicator you can do whatever you want: reduce it, split it into parts, raise it to another power, etc.

Degree with irrational exponent

Let α – irrational number, and A ˃ 0.

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

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

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

• 0˂А˂1.

In this case, it’s the other way around: A r 2 ˂ A α ˂ A r 1 under the same conditions as in the second paragraph.

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

r 1 – in this case equals 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 quantities needed for, what are the advantages of such functions? Of course, first of all, they simplify the life of mathematicians and programmers when solving examples, since they allow them to minimize calculations, shorten algorithms, systematize data, and much more.

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

I. Work n factors, each of which is equal A called n-th power of the number A and is designated An.

Examples. Write the product as a degree.

1) mmmm; 2) aaabb; 3) 5 5 5 5 ccc; 4) ppkk+pppk-ppkkk.

Solution.

1) mmmm=m 4, since, by definition of a degree, the product of four factors, each of which is equal m, will fourth power of m.

2) aaabb=a 3 b 2 ; 3) 5·5·5·5·ccc=5 4 c 3 ; 4) ppkk+pppk-ppkkk=p 2 k 2 +p 3 k-p 2 k 3.

II. The action by which the product of several equal factors is found is called exponentiation. The number that is raised to a power is called the base of the power. The number that shows to what power the base is raised is called the exponent. So, An- degree, A– the basis of the degree, n– exponent. For example:

2 3 — it's a degree. Number 2 is the base of the degree, the exponent is equal to 3 . Degree value 2 3 equals 8, because 2 3 =2·2·2=8.

Examples. Write the following expressions without the exponent.

5) 4 3; 6) a 3 b 2 c 3 ; 7) a 3 -b 3 ; 8) 2a 4 +3b 2 .

Solution.

5) 4 3 = 4·4·4 ; 6) a 3 b 2 c 3 = aaabbccc; 7) a 3 -b 3 = aaa-bbb; 8) 2a 4 +3b 2 = 2aaaa+3bb.

III. and 0 =1 Any number (except zero) to the zero power is equal to one. For example, 25 0 =1.
IV. a 1 =aAny number to the first power is equal to itself.

V. a m∙ a n= a m + n When multiplying powers with the same bases, the base is left the same, and the exponents folded

Examples. Simplify:

9) a·a 3 ·a 7 ; 10) b 0 +b 2 b 3 ; 11) c 2 ·c 0 ·c·c 4 .

Solution.

9) a·a 3 ·a 7=a 1+3+7 =a 11 ; 10) b 0 +b 2 b 3 = 1+b 2+3 =1+b 5 ;

11) c 2 c 0 c c 4 = 1 c 2 c c 4 =c 2+1+4 =c 7 .

VI. a m: a n= a m - nWhen dividing powers with the same base, the base is left the same, and the exponent of the divisor is subtracted from the exponent of the dividend.

Examples. Simplify:

12) a 8:a 3 ; 13) m 11:m 4 ; 14) 5 6:5 4 .

12)a 8:a 3=a 8-3 =a 5 ; 13)m 11:m 4=m 11-4 =m 7; 14 ) 5 6:5 4 =5 2 =5·5=25.

VII. (a m) n= a mn When raising a power to a power, the base is left the same, and the exponents are multiplied.

Examples. Simplify:

15) (a 3) 4 ; 16) (c 5) 2.

15) (a 3) 4=a 3·4 =a 12 ; 16) (c 5) 2=c 5 2 =c 10.

Please note, which, since the product does not change from rearranging the factors, That:

15) (a 3) 4 = (a 4) 3 ; 16) (c 5) 2 = (c 2) 5 .

VI II. (a∙b) n =a n ∙b n When raising a product to a power, each of the factors is raised to that power.

Examples. Simplify:

17) (2a 2) 5 ; 18) 0.2 6 ·5 6 ; 19) 0.25 2 40 2.

Solution.

17) (2a 2) 5=2 5 ·a 2·5 =32a 10 ; 18) 0.2 6 5 6=(0.2·5) 6 =1 6 =1;

19) 0.25 2 40 2=(0.25·40) 2 =10 2 =100.

IX. When raising a fraction to a power, both the numerator and denominator of the fraction are raised to that power.

Examples. Simplify:

Solution.

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Lesson type: lesson of generalization and systematization of knowledge

Goals:

• educational– repeat the definition of a degree, the rules for multiplying and dividing degrees, raising a degree to a power, consolidate the skills of solving examples containing degrees,
• developing- development logical thinking students, interest in the material being studied,
• raising– fostering a responsible attitude to learning, a culture of communication, and a sense of collectivism.

Equipment: computer, multimedia projector, interactive whiteboard, presentation of “Degrees” for mental calculation, cards with tasks, handouts.

Lesson plan:

1. Organizational moment.
2. Repetition of rules
3. Oral counting.
4. Historical information.
5. Work at the board.
6. Physical education minute.
7. Working on an interactive whiteboard.
8. Independent work.
9. Homework.
10. Summing up the lesson.

Lesson progress

I. Organizational moment

Communicate the topic and objectives of the lesson.

In previous lessons you discovered amazing world degrees, learned to multiply and divide degrees, and raise them to a power. Today we must consolidate the acquired knowledge by solving examples.

II. Repetition of rules(orally)

1. Give the definition of degree with a natural exponent? (Power of number A with a natural exponent greater than 1 is called a product n factors, each of which is equal A.)
2. How to multiply two powers? (To multiply powers with the same bases, you must leave the base the same and add the exponents.)
3. How to divide degree by degree? (To divide powers with the same bases, you need to leave the base the same and subtract the exponents.)
4. How to raise a product to a power? (To raise a product to a power, you need to raise each factor to that power)
5. How to raise a degree to a power? (To raise a power to a power, you need to leave the base the same and multiply the exponents)

III. Oral counting(by multimedia)

IV. Historical background

All problems are from the Ahmes papyrus, which was written around 1650 BC. e. associated with construction practice, delimitation land plots etc. Tasks are grouped by topic. These are mainly tasks on finding the areas of a triangle, quadrilaterals and a circle, various operations with integers and fractions, proportional division, finding ratios, there is also raising in different degrees, solving equations of the first and second degree with one unknown.

There is a complete lack of any explanation or evidence. The desired result is either given directly or given short algorithm its calculations. This method of presentation, typical for science in countries ancient East, suggests that mathematics there developed through generalizations and guesses that did not form any general theory. However, the papyrus contains a number of evidence that Egyptian mathematicians knew how to extract roots and raise to powers, solve equations, and even mastered the rudiments of algebra.

V. Work at the board

Find the meaning of the expression in a rational way:

Calculate the value of the expression:

VI. Physical education minute

1. for the eyes
2. for the neck
3. for hands
4. for the torso
5. for feet

VII. Problem solving(with display on the interactive whiteboard)

Is the root of the equation a positive number?

a) 3x + (-0.1) 7 = (-0.496) 4 (x > 0)

b) (10.381) 5 = (-0.012) 3 - 2x (x< 0)

VIII. Independent work

IX. Homework

X. Summing up the lesson

Analysis of results, announcement of grades.

We will use the acquired knowledge about degrees when solving equations and problems in high school; they are also often found in the Unified State Exam.

It is obvious that numbers with powers can be added like other quantities , by adding them one after another with their signs.

So, the sum of a 3 and b 2 is a 3 + b 2.
The sum of a 3 - b n and h 5 -d 4 is a 3 - b n + h 5 - d 4.

Odds equal degrees identical variables can be added or subtracted.

So, the sum of 2a 2 and 3a 2 is equal to 5a 2.

It is also obvious that if you take two squares a, or three squares a, or five squares a.

But degrees various variables And various degrees identical variables, must be composed by adding them with their signs.

So, the sum of a 2 and a 3 is the sum of a 2 + a 3.

It is obvious that the square of a, and the cube of a, is not equal to twice the square of a, but to twice the cube of a.

The sum of a 3 b n and 3a 5 b 6 is a 3 b n + 3a 5 b 6.

Subtraction powers are carried out in the same way as addition, except that the signs of the subtrahends must be changed accordingly.

Or:
2a 4 - (-6a 4) = 8a 4
3h 2 b 6 - 4h 2 b 6 = -h 2 b 6
5(a - h) 6 - 2(a - h) 6 = 3(a - h) 6

Multiplying powers

Numbers with powers can be multiplied, like other quantities, by writing them one after the other, with or without a multiplication sign between them.

Thus, the result of multiplying a 3 by b 2 is a 3 b 2 or aaabb.

Or:
x -3 ⋅ a m = a m x -3
3a 6 y 2 ⋅ (-2x) = -6a 6 xy 2
a 2 b 3 y 2 ⋅ a 3 b 2 y = a 2 b 3 y 2 a 3 b 2 y

The result in the last example can be ordered by adding identical variables.
The expression will take the form: a 5 b 5 y 3.

By comparing several numbers (variables) with powers, we can see that if any two of them are multiplied, then the result is a number (variable) with a power equal to amount degrees of terms.

So, a 2 .a 3 = aa.aaa = aaaaa = a 5 .

Here 5 is the power of the result of the multiplication, equal to 2 + 3, the sum of the powers of the terms.

So, a n .a m = a m+n .

For a n , a is taken as a factor as many times as the power of n;

And a m is taken as a factor as many times as the degree m is equal to;

That's why, powers with the same bases can be multiplied by adding the exponents of the powers.

So, a 2 .a 6 = a 2+6 = a 8 . And x 3 .x 2 .x = x 3+2+1 = x 6 .

Or:
4a n ⋅ 2a n = 8a 2n
b 2 y 3 ⋅ b 4 y = b 6 y 4
(b + h - y) n ⋅ (b + h - y) = (b + h - y) n+1

Multiply (x 3 + x 2 y + xy 2 + y 3) ⋅ (x - y).
Answer: x 4 - y 4.
Multiply (x 3 + x - 5) ⋅ (2x 3 + x + 1).

This rule is also true for numbers whose exponents are negative.

1. So, a -2 .a -3 = a -5 . This can be written as (1/aa).(1/aaa) = 1/aaaaa.

2. y -n .y -m = y -n-m .

3. a -n .a m = a m-n .

If a + b are multiplied by a - b, the result will be a 2 - b 2: that is

The result of multiplying the sum or difference of two numbers is equal to the sum or difference of their squares.

If you multiply the sum and difference of two numbers raised to square, the result will be equal to the sum or difference of these numbers in fourth degrees.

So, (a - y).(a + y) = a 2 - y 2.
(a 2 - y 2)⋅(a 2 + y 2) = a 4 - y 4.
(a 4 - y 4)⋅(a 4 + y 4) = a 8 - y 8.

Division of degrees

Numbers with powers can be divided like other numbers, by subtracting from the dividend, or by placing them in fraction form.

Thus, a 3 b 2 divided by b 2 is equal to a 3.

Or:
$\frac(9a^3y^4)(-3a^3) = -3y^4$
$\frac(a^2b + 3a^2)(a^2) = \frac(a^2(b+3))(a^2) = b + 3$
$\frac(d\cdot (a - h + y)^3)((a - h + y)^3) = d$

Writing a 5 divided by a 3 looks like $\frac(a^5)(a^3)$. But this is equal to a 2 . In a series of numbers
a +4 , a +3 , a +2 , a +1 , a 0 , a -1 , a -2 , a -3 , a -4 .
any number can be divided by another, and the exponent will be equal to difference indicators of divisible numbers.

When dividing degrees with the same basis their indicators are subtracted..

So, y 3:y 2 = y 3-2 = y 1. That is, $\frac(yyy)(yy) = y$.

And a n+1:a = a n+1-1 = a n . That is, $\frac(aa^n)(a) = a^n$.

Or:
y 2m: y m = y m
8a n+m: 4a m = 2a n
12(b + y) n: 3(b + y) 3 = 4(b +y) n-3

The rule is also true for numbers with negative values ​​of degrees.
The result of dividing a -5 by a -3 is a -2.
Also, $\frac(1)(aaaaa) : \frac(1)(aaa) = \frac(1)(aaaaa).\frac(aaa)(1) = \frac(aaa)(aaaaa) = \frac (1)(aa)$.

h 2:h -1 = h 2+1 = h 3 or $h^2:\frac(1)(h) = h^2.\frac(h)(1) = h^3$

It is necessary to master multiplication and division of powers very well, since such operations are very widely used in algebra.

Examples of solving examples with fractions containing numbers with powers

1. Reduce the exponents by $\frac(5a^4)(3a^2)$ Answer: $\frac(5a^2)(3)$.

2. Decrease the exponents by $\frac(6x^6)(3x^5)$. Answer: $\frac(2x)(1)$ or 2x.

3. Reduce the exponents a 2 /a 3 and a -3 /a -4 and bring to a common denominator.
a 2 .a -4 is a -2 the first numerator.
a 3 .a -3 is a 0 = 1, the second numerator.
a 3 .a -4 is a -1 , the common numerator.
After simplification: a -2 /a -1 and 1/a -1 .

4. Reduce the exponents 2a 4 /5a 3 and 2 /a 4 and bring to a common denominator.
Answer: 2a 3 /5a 7 and 5a 5 /5a 7 or 2a 3 /5a 2 and 5/5a 2.

5. Multiply (a 3 + b)/b 4 by (a - b)/3.

6. Multiply (a 5 + 1)/x 2 by (b 2 - 1)/(x + a).

7. Multiply b 4 /a -2 by h -3 /x and a n /y -3 .

8. Divide a 4 /y 3 by a 3 /y 2 . Answer: a/y.

9. Divide (h 3 - 1)/d 4 by (d n + 1)/h.

Lesson on the topic: "Rules of multiplication and division of powers with the same and different exponents. Examples"

Additional materials
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Educational aids and simulators in the Integral online store for grade 7
Manual for the textbook Yu.N. Makarycheva Manual for the textbook by A.G. Mordkovich

Purpose of the lesson: learn to perform operations with powers of numbers.

First, let's remember the concept of "power of number". An expression of the form $\underbrace( a * a * \ldots * a )_(n)$ can be represented as $a^n$.

The converse is also true: $a^n= \underbrace( a * a * \ldots * a )_(n)$.

This equality is called “recording the degree as a product.” It will help us determine how to multiply and divide powers.
Remember:
a– the basis of the degree.
n– exponent.
If n=1, which means the number A took once and accordingly: $a^n= 1$.
If n= 0, then $a^0= 1$.

We can find out why this happens when we get acquainted with the rules of multiplication and division of powers.

Multiplication rules

a) If powers with the same base are multiplied.
To get $a^n * a^m$, we write the degrees as a product: $\underbrace( a * a * \ldots * a )_(n) * \underbrace( a * a * \ldots * a )_(m )$.
The figure shows that the number A took n+m times, then $a^n * a^m = a^(n + m)$.

Example.
$2^3 * 2^2 = 2^5 = 32$.

This property is convenient to use to simplify the work when raising a number to a higher power.
Example.
$2^7= 2^3 * 2^4 = 8 * 16 = 128$.

b) If powers are multiplied with different reasons, but with the same indicator.
To get $a^n * b^n$, we write the degrees as a product: $\underbrace( a * a * \ldots * a )_(n) * \underbrace( b * b * \ldots * b )_(m )$.
If we swap the factors and count the resulting pairs, we get: $\underbrace( (a * b) * (a * b) * \ldots * (a * b) )_(n)$.

So $a^n * b^n= (a * b)^n$.

Example.
$3^2 * 2^2 = (3 * 2)^2 = 6^2= 36$.

Division rules

a) The basis of the degree is the same, the indicators are different.
Consider dividing a power with a larger exponent by dividing a power with a smaller exponent.

So, we need $\frac(a^n)(a^m)$, Where n>m.

Let's write the degrees as a fraction:

$\frac(\underbrace( a * a * \ldots * a )_(n))(\underbrace( a * a * \ldots * a )_(m))$.

For convenience, we write the division as a simple fraction.

Now let's reduce the fraction.

It turns out: $\underbrace( a * a * \ldots * a )_(n-m)= a^(n-m)$.
Means, $\frac(a^n)(a^m)=a^(n-m)$.

This property will help explain the situation with raising a number to the zero power. Let's assume that n=m, then $a^0= a^(n-n)=\frac(a^n)(a^n) =1$.

Examples.
$\frac(3^3)(3^2)=3^(3-2)=3^1=3$.

$\frac(2^2)(2^2)=2^(2-2)=2^0=1$.

b) The bases of the degree are different, the indicators are the same.
Let's say we need $\frac(a^n)( b^n)$. Let's write powers of numbers as fractions:

$\frac(\underbrace( a * a * \ldots * a )_(n))(\underbrace( b * b * \ldots * b )_(n))$.

For convenience, let's imagine.

Using the property of fractions, we divide the large fraction into the product of small ones, we get.
$\underbrace( \frac(a)(b) * \frac(a)(b) * \ldots * \frac(a)(b) )_(n)$.
Accordingly: $\frac(a^n)( b^n)=(\frac(a)(b))^n$.

Example.
$\frac(4^3)( 2^3)= (\frac(4)(2))^3=2^3=8$.