We remind you that in this lesson we will understand properties of degrees with natural indicators and zero.

Powers with rational exponents and their properties will be discussed in lessons for 8th grade.

A power with a natural exponent has several important properties that allow us to simplify calculations in examples with powers.
Property No. 1

Product of powers

Remember!

When multiplying powers with the same bases, the base remains unchanged, and the exponents of the powers are added.

a m · a n = a m + n, where “a” is any number, and “m”, “n” are any natural numbers.

• This property of powers also applies to the product of three or more powers.
  Simplify the expression.
• b b 2 b 3 b 4 b 5 = b 1 + 2 + 3 + 4 + 5 = b 15
  Present it as a degree.
• b b 2 b 3 b 4 b 5 = b 1 + 2 + 3 + 4 + 5 = b 15
  6 15 36 = 6 15 6 2 = 6 15 6 2 = 6 17

(0.8) 3 · (0.8) 12 = (0.8) 3 + 12 = (0.8) 15

Important! Please note that in the indicated property we were only talking about multiplying powers with on the same grounds

. It does not apply to their addition.
You cannot replace the sum (3 3 + 3 2) with 3 5. This is understandable if

calculate (3 3 + 3 2) = (27 + 9) = 36, and 3 5 = 243
Property No. 2

Product of powers

Partial degrees

When dividing powers with the same bases, the base remains unchanged, and the exponent of the divisor is subtracted from the exponent of the dividend.

= 11 3 − 2 4 2 − 1 = 11 4 = 44
Example. Solve the equation. We use the property of quotient powers.

3 8: t = 3 4

T = 3 8 − 4

Answer: t = 3 4 = 81

• Using properties No. 1 and No. 2, you can easily simplify expressions and perform calculations.
  Example. Simplify the expression.
• 4 5m + 6 4 m + 2: 4 4m + 3 = 4 5m + 6 + m + 2: 4 4m + 3 = 4 6m + 8 − 4m − 3 = 4 2m + 5
  = = = 2 9 + 2

  2 5

  = 2 11

  2 5

  = 2 11 − 5 = 2 6 = 64

  (0.8) 3 · (0.8) 12 = (0.8) 3 + 12 = (0.8) 15

  Example. Find the value of an expression using the properties of exponents.

  Please note that in Property 2 we were only talking about dividing powers with the same bases. (4 3 −4 2) = (64 − 16) = 48 You cannot replace the difference (4 3 −4 2) with 4 1. This is understandable if you count

  , and 4 1 = 4

  Be careful!
  Property No. 3

  Product of powers

  Raising a degree to a power

  (a n) m = a n · m, where “a” is any number, and “m”, “n” are any natural numbers.

  
  

  Properties 4
  Product power

  Product of powers

  When raising a product to a power, each of the factors is raised to a power. The results obtained are then multiplied.

  (a b) n = a n b n, where “a”, “b” are any rational numbers; "n" is any natural number.

  • Example 1.
    (6 a 2 b 3 c) 2 = 6 2 a 2 2 b 3 2 c 1 2 = 36 a 4 b 6 c 2
    
  • Example 2.
    (−x 2 y) 6 = ((−1) 6 x 2 6 y 1 6) = x 12 y 6

  (0.8) 3 · (0.8) 12 = (0.8) 3 + 12 = (0.8) 15

  Please note that property No. 4, like other properties of degrees, is also applied in reverse order.

  (a n b n)= (a b) n

  That is, to multiply powers with the same exponents, you can multiply the bases, but leave the exponent unchanged.

  • Example. Calculate.
    2 4 5 4 = (2 5) 4 = 10 4 = 10,000
  • Example. Calculate.
    0.5 16 2 16 = (0.5 2) 16 = 1

  In more complex examples, there may be cases where multiplication and division must be performed over powers with different bases and different exponents.

  In this case, we advise you to do the following. For example,
  

  4 5 3 2 = 4 3 4 2 3 2 = 4 3 (4 3) 2 = 64 12 2 = 64 144 = 9216

  An example of raising a decimal to a power.

  4 21 (−0.25) 20 = 4 4 20 (−0.25) 20 = 4 (4 (−0.25)) 20 = 4 (−1) 20 = 4 1 = 4
  Properties 5

  Product of powers

  Power of a quotient (fraction)

  To raise a quotient to a power, you can raise the dividend and the divisor separately to this power, and divide the first result by the second.

  • (a: b) n = a n: b n, where “a”, “b” are any rational numbers, b ≠ 0, n is any natural number.
    (5: 3) 12 = 5 12: 3 12

  Example. Present the expression as a quotient of powers.

We remind you that a quotient can be represented as a fraction. Therefore, we will dwell on the topic of raising a fraction to a power in more detail on the next page. 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 powers of 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.

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 base, their exponents 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.

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 consider the most important issues regarding this definition. Namely, let’s 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 raising 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 complex 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 raising the product of 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 is equal to the product of the numerator and the denominator to a positive power.
6. Any number to the power 0 = 1, and to the power. 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 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 type can be written as a scheme: A m / n. Read as: the nth root of the number A to the power m.

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

Degree with irrational exponent

Let α be an 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.

If you need to raise a specific number to a power, you can use . Now we will take a closer look at properties of degrees.

Exponential numbers open up great possibilities, they allow us to transform multiplication into addition, and adding is much easier than multiplying.

For example, we need to multiply 16 by 64. The product of multiplying these two numbers is 1024. But 16 is 4x4, and 64 is 4x4x4. That is, 16 by 64 = 4x4x4x4x4, which is also equal to 1024.

The number 16 can also be represented as 2x2x2x2, and 64 as 2x2x2x2x2x2, and if we multiply, we again get 1024.

Now let's use the rule. 16=4 2, or 2 4, 64=4 3, or 2 6, at the same time 1024=6 4 =4 5, or 2 10.

Therefore, our problem can be written differently: 4 2 x4 3 =4 5 or 2 4 x2 6 =2 10, and each time we get 1024.

We can solve a number of similar examples and see that multiplying numbers with powers reduces to adding exponents, or exponential, of course, provided that the bases of the factors are equal.

Thus, without performing multiplication, we can immediately say that 2 4 x2 2 x2 14 = 2 20.

This rule is also valid when dividing numbers with powers, but in this case the exponent of the divisor is subtracted from the exponent of the dividend. Thus, 2 5:2 3 =2 2, which in ordinary numbers is equal to 32:8 = 4, that is, 2 2. Let's summarize:

a m x a n =a m+n, a m: a n =a m-n, where m and n are integers.

At first glance it may seem that this is multiplying and dividing numbers with powers not very convenient, because first you need to represent the number in exponential form. It is not difficult to represent the numbers 8 and 16, that is, 2 3 and 2 4, in this form, but how to do this with the numbers 7 and 17? Or what to do in cases where a number can be represented in exponential form, but the bases for exponential expressions of numbers are very different. For example, 8x9 is 2 3 x 3 2, in which case we cannot sum the exponents. Neither 2 5 nor 3 5 are the answer, nor does the answer lie in the interval between these two numbers.

Then is it worth bothering with this method at all? Definitely worth it. It provides enormous benefits, especially for complex and time-consuming calculations.

Additional materials
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Teaching 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 have taken 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 with different bases but the same exponent are multiplied.
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.
$\frac(4^3)( 2^3)= (\frac(4)(2))^3=2^3=8$.

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 consider the most important issues regarding this definition. Namely, let’s 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 raising 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 complex 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 raising the product of 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 is equal to the product of the numerator and the denominator to a positive power.
6. Any number to the power 0 = 1, and to the power. 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 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 type can be written as a scheme: A m / n. Read as: the nth root of the number A to the power m.

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

Degree with irrational exponent

Let α be an 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.

Expressions, expression conversion

Power expressions (expressions with powers) and their transformation

In this article we will talk about converting expressions with powers. First, we will focus on transformations that are performed with expressions of any kind, including power expressions, such as opening parentheses and bringing similar terms. And then we will analyze the transformations inherent specifically in expressions with degrees: working with the base and exponent, using the properties of degrees, etc.

Page navigation.

What are power expressions?

The term “power expressions” practically does not appear in school mathematics textbooks, but it appears quite often in collections of problems, especially those intended for preparation for the Unified State Exam and the Unified State Exam, for example. After analyzing the tasks in which it is necessary to perform any actions with power expressions, it becomes clear that power expressions are understood as expressions containing powers in their entries. Therefore, you can accept the following definition for yourself:

Definition.

Power expressions are expressions containing degrees.

Let's give examples of power expressions. Moreover, we will present them according to how the development of views on from a degree with a natural exponent to a degree with a real exponent occurs.

As is known, first one gets acquainted with the power of a number with a natural exponent; at this stage, the first simplest power expressions of the type 3 2, 7 5 +1, (2+1) 5, (−0.1) 4, 3 a 2 appear −a+a 2 , x 3−1 , (a 2) 3 etc.

A little later, the power of a number with an integer exponent is studied, which leads to the appearance of power expressions with negative integer powers, like the following: 3 −2, , a −2 +2 b −3 +c 2 .

In high school they return to degrees. There a degree with a rational exponent is introduced, which entails the appearance of the corresponding power expressions: , and so on. Finally, degrees with irrational exponents and expressions containing them are considered: , .

The matter is not limited to the listed power expressions: further the variable penetrates into the exponent, and, for example, the following expressions arise: 2 x 2 +1 or . And after getting acquainted with, expressions with powers and logarithms begin to appear, for example, x 2·lgx −5·x lgx.

So, we have dealt with the question of what power expressions represent. Next we will learn to transform them.

Main types of transformations of power expressions

With power expressions, you can perform any of the basic identity transformations of expressions. For example, you can open parentheses, replace numerical expressions with their values, add similar terms, etc. Naturally, it is necessary to follow the accepted procedure for performing actions. Let's give examples.

Calculate the value of the power expression 2 3 ·(4 2 −12) .

According to the order of execution of actions, first perform the actions in brackets. There, firstly, we replace the power 4 2 with its value 16 (see if necessary), and secondly, we calculate the difference 16−12=4. We have 2 3 ·(4 2 −12)=2 3 ·(16−12)=2 3 ·4.

In the resulting expression, we replace the power 2 3 with its value 8, after which we calculate the product 8·4=32. This is the desired value.

So, 2 3 ·(4 2 −12)=2 3 ·(16−12)=2 3 ·4=8·4=32.

2 3 ·(4 2 −12)=32.

Simplify expressions with powers 3 a 4 b −7 −1+2 a 4 b −7.

Obviously, this expression contains similar terms 3·a 4 ·b −7 and 2·a 4 ·b −7 , and we can present them: .

3·a 4 ·b −7 −1+2·a 4 ·b −7 =5·a 4 ·b −7 −1 .

Express an expression with powers as a product.

You can cope with the task by representing the number 9 as a power of 3 2 and then using the formula for abbreviated multiplication - difference of squares:

There are also a number of identical transformations inherent specifically in power expressions. We will analyze them further.

Working with base and exponent

There are degrees whose base and/or exponent are not just numbers or variables, but some expressions. As an example, we give the entries (2+0.3·7) 5−3.7 and (a·(a+1)−a 2) 2·(x+1) .

When working with such expressions, you can replace both the expression in the base of the degree and the expression in the exponent with an identically equal expression in the ODZ of its variables. In other words, according to the rules known to us, we can separately transform the base of the degree and separately the exponent. It is clear that as a result of this transformation, an expression will be obtained that is identically equal to the original one.

Such transformations allow us to simplify expressions with powers or achieve other goals we need. For example, in the power expression mentioned above (2+0.3 7) 5−3.7, you can perform operations with the numbers in the base and exponent, which will allow you to move to the power 4.1 1.3. And after opening the brackets and bringing similar terms to the base of the degree (a·(a+1)−a 2) 2·(x+1), we obtain a power expression of a simpler form a 2·(x+1) .

Using Degree Properties

One of the main tools for transforming expressions with powers is equalities, reflecting. Let us recall the main ones. For any positive numbers a and b and arbitrary real numbers r and s, the following properties of powers are true:

• a r ·a s =a r+s ;
• a r:a s =a r−s ;
• (a·b) r =a r ·b r ;
• (a:b) r =a r:b r ;
• (a r) s =a r·s .

Note that for natural, integer, and positive exponents, the restrictions on the numbers a and b may not be so strict. For example, for natural numbers m and n the equality a m ·a n =a m+n is true not only for positive a, but also for negative a, and for a=0.

At school, the main focus when transforming power expressions is on the ability to choose the appropriate property and apply it correctly. In this case, the bases of degrees are usually positive, which allows the properties of degrees to be used without restrictions. The same applies to the transformation of expressions containing variables in the bases of powers - the range of permissible values ​​of variables is usually such that the bases take only positive values ​​on it, which allows you to freely use the properties of powers. In general, you need to constantly ask yourself whether it is possible to use any property of degrees in this case, because inaccurate use of properties can lead to a narrowing of the educational value and other troubles. These points are discussed in detail and with examples in the article transformation of expressions using properties of degrees. Here we will limit ourselves to considering a few simple examples.

Express the expression a 2.5 ·(a 2) −3:a −5.5 as a power with base a.

First, we transform the second factor (a 2) −3 using the property of raising a power to a power: (a 2) −3 =a 2·(−3) =a −6. The original power expression will take the form a 2.5 ·a −6:a −5.5. Obviously, it remains to use the properties of multiplication and division of powers with the same base, we have
a 2.5 ·a −6:a −5.5 =
a 2.5−6:a −5.5 =a −3.5:a −5.5 =
a −3.5−(−5.5) =a 2 .

a 2.5 ·(a 2) −3:a −5.5 =a 2.

Properties of powers when transforming power expressions are used both from left to right and from right to left.

Find the value of the power expression.

The equality (a·b) r =a r ·b r , applied from right to left, allows us to move from the original expression to a product of the form and further. And when multiplying powers with the same bases, the exponents add up: .

It was possible to transform the original expression in another way:

.

Given the power expression a 1.5 −a 0.5 −6, introduce a new variable t=a 0.5.

The degree a 1.5 can be represented as a 0.5 3 and then, based on the property of the degree to the degree (a r) s =a r s, applied from right to left, transform it to the form (a 0.5) 3. Thus, a 1.5 −a 0.5 −6=(a 0.5) 3 −a 0.5 −6. Now it’s easy to introduce a new variable t=a 0.5, we get t 3 −t−6.

Converting fractions containing powers

Power expressions can contain or represent fractions with powers. Any of the basic transformations of fractions that are inherent in fractions of any kind are fully applicable to such fractions. That is, fractions that contain powers can be reduced, reduced to a new denominator, worked separately with their numerator and separately with the denominator, etc. To illustrate these words, consider solutions to several examples.

Simplify power expression .

This power expression is a fraction. Let's work with its numerator and denominator. In the numerator we open the brackets and simplify the resulting expression using the properties of powers, and in the denominator we present similar terms:

And let’s also change the sign of the denominator by placing a minus in front of the fraction: .

.

Reducing fractions containing powers to a new denominator is carried out similarly to reducing rational fractions to a new denominator. In this case, an additional factor is also found and the numerator and denominator of the fraction are multiplied by it. When performing this action, it is worth remembering that reduction to a new denominator can lead to a narrowing of the VA. To prevent this from happening, it is necessary that the additional factor does not go to zero for any values ​​of the variables from the ODZ variables for the original expression.

Reduce the fractions to a new denominator: a) to denominator a, b) to the denominator.

a) In this case, it is quite easy to figure out which additional multiplier helps to achieve the desired result. This is a multiplier of a 0.3, since a 0.7 ·a 0.3 =a 0.7+0.3 =a. Note that in the range of permissible values ​​of the variable a (this is the set of all positive real numbers), the power of a 0.3 does not vanish, therefore, we have the right to multiply the numerator and denominator of a given fraction by this additional factor:

b) Taking a closer look at the denominator, you will find that

and multiplying this expression by will give the sum of cubes and, that is, . And this is the new denominator to which we need to reduce the original fraction.

This is how we found an additional multiplier. In the range of permissible values ​​of the variables x and y, the expression does not vanish, therefore, we can multiply the numerator and denominator of the fraction by it:

A) , b) .

There is also nothing new in reducing fractions containing powers: the numerator and denominator are represented as a number of factors, and the same factors of the numerator and denominator are reduced.

Reduce the fraction: a) , b) .

a) Firstly, the numerator and denominator can be reduced by the numbers 30 and 45, which is equal to 15. It is also obviously possible to perform a reduction by x 0.5 +1 and by . Here's what we have:

b) In this case, identical factors in the numerator and denominator are not immediately visible. To obtain them, you will have to perform preliminary transformations. In this case, they consist in factoring the denominator using the difference of squares formula:

A)

b) .

Converting fractions to a new denominator and reducing fractions are mainly used to do things with fractions. Actions are performed according to known rules. When adding (subtracting) fractions, they are reduced to a common denominator, after which the numerators are added (subtracted), but the denominator remains the same. The result is a fraction whose numerator is the product of the numerators, and the denominator is the product of the denominators. Division by a fraction is multiplication by its inverse.

Follow the steps .

First, we subtract the fractions in parentheses. To do this, we bring them to a common denominator, which is , after which we subtract the numerators:

Now we multiply the fractions:

Obviously, it is possible to reduce by a power of x 1/2, after which we have .

You can also simplify the power expression in the denominator by using the difference of squares formula: .

Simplify the Power Expression .

Obviously, this fraction can be reduced by (x 2.7 +1) 2, this gives the fraction . It is clear that something else needs to be done with the powers of X. To do this, transform the resulting fraction into a product. This gives us the opportunity to take advantage of the property of dividing powers with the same bases: . And at the end of the process we move from the last product to the fraction.

.

And let us also add that it is possible, and in many cases desirable, to transfer factors with negative exponents from the numerator to the denominator or from the denominator to the numerator, changing the sign of the exponent. Such transformations often simplify further actions. For example, the power expression can be replaced by.

Converting expressions with roots and powers

Often, in expressions in which some transformations are required, roots with fractional exponents are also present along with powers. To transform such an expression to the desired form, in most cases it is enough to go only to roots or only to powers. But since it is more convenient to work with powers, they usually move from roots to powers. However, it is advisable to carry out such a transition when the ODZ of variables for the original expression allows you to replace the roots with powers without the need to refer to the module or split the ODZ into several intervals (we discussed this in detail in the article transition from roots to powers and back After getting acquainted with the degree with a rational exponent a degree with an irrational exponent is introduced, which allows us to talk about a degree with an arbitrary real exponent. At this stage, it begins to be studied at school. exponential function, which is analytically given by a power, the base of which is a number, and the exponent is a variable. So we are faced with power expressions containing numbers in the base of the power, and in the exponent - expressions with variables, and naturally the need arises to perform transformations of such expressions.

It should be said that the transformation of expressions of the indicated type usually has to be performed when solving exponential equations various variables exponential inequalities, and these conversions are quite simple. In the overwhelming majority of cases, they are based on the properties of the degree and are aimed, for the most part, at introducing a new variable in the future. The equation will allow us to demonstrate them 5 2 x+1 −3 5 x 7 x −14 7 2 x−1 =0.

Firstly, powers, in the exponents of which is the sum of a certain variable (or expression with variables) and a number, are replaced by products. This applies to the first and last terms of the expression on the left side:
5 2 x 5 1 −3 5 x 7 x −14 7 2 x 7 −1 =0,
5 5 2 x −3 5 x 7 x −2 7 2 x =0.

Next, both sides of the equality are divided by the expression 7 2 x, which on the ODZ of the variable x for the original equation takes only positive values ​​(this is a standard technique for solving equations of this type, we are not talking about it now, so focus on subsequent transformations of expressions with powers ):

Now we can cancel fractions with powers, which gives .

Finally, the ratio of powers with the same exponents is replaced by powers of relations, resulting in the equation , which is equivalent . The transformations made allow us to introduce a new variable, which reduces the solution of the original exponential equation to the solution of a quadratic equation

I. V. Boykov, L. D. Romanova Collection of tasks for preparing for the Unified State Exam. Part 1. Penza 2003.

Sections: Mathematics

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 of students’ logical thinking, 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, task cards, handouts.

Lesson plan:

Organizing time.

Repetition of rules

Verbal counting.

Historical reference.

Work at the board.

Physical education minute.

Working on an interactive whiteboard.

Independent work.

Homework.

Summing up the lesson.

During the classes

I. Organizational moment

Communicate the topic and objectives of the lesson.

In previous lessons, you discovered the wonderful world of powers, learned how to multiply and divide powers, and raise them to powers. 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. Verbal counting(by multimedia)

IV. Historical reference

All problems are from the Ahmes papyrus, which was written around 1650 BC. e. related to construction practice, demarcation of 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 to different powers, 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 a short algorithm for calculating it is given. This method of presentation, typical of science in the countries of the 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

6. for eyes
7. for the neck
8. for hands
9. for the torso
10. for legs

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

Is the root of the equation a positive number?

xn--i1abbnckbmcl9fb.xn--p1ai

Formulas of powers and roots.

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

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



Operations with degrees.

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

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



3. The degree of the product of 2 or more factors is equal to the product of the degrees of these factors:

(abc…) n = a n · b n · c n …

4. The degree of a fraction is equal to the ratio of the degrees of the dividend and the divisor:

5. Raising a power to a power, the exponents are multiplied:

Each formula above is true in the directions from left to right and vice versa.

Operations with roots.

1. The root of the product of several factors is equal to the product of the roots of these factors:

2. The root of a ratio is equal to the ratio of the dividend and the divisor of the roots:



3. When raising a root to a power, it is enough to raise the radical number to this power:



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



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



The power of a certain number with a non-positive (integer) exponent is defined as one divided by the power of the same number with an exponent equal to the absolute value of the non-positive exponent:



Formula a m :a n =a m - n can be used not only for m > n, but also with m 4:a 7 = a 4 - 7 = a -3 .

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

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

To raise a real number A to the degree m/n, you need to extract the root n-th degree from m-th power of this number A:



Degree formulas.

6. a - n = - division of degrees;

7. - division of degrees;

8. a 1/n = ;

Degrees of rules of action with degrees

1. The degree of the product of two or more factors is equal to the product of the degrees of these factors (with the same exponent):

(abc…) n = a n b n c n …

Example 1. (7 2 10) 2 = 7 2 2 2 10 2 = 49 4 100 = 19600. Example 2. (x 2 –a 2) 3 = [(x +a)(x - a)] 3 =( x +a) 3 (x - a) 3

In practice, the reverse conversion is more important:

a n b n c n … = (abc…) n

those. the product of identical powers of several quantities is equal to the same power of the product of these quantities.

Example 3. Example 4. (a +b) 2 (a 2 – ab +b 2) 2 =[(a +b)(a 2 – ab +b 2)] 2 =(a 3 +b 3) 2

2. The power of a quotient (fraction) is equal to the quotient of dividing the same power of the divisor by the same power:

Example 5. Example 6.

Reverse conversion:. Example 7. . Example 8. .

3. When multiplying degrees with the same bases, the exponents of the degrees are added:

Example 9.2 2 2 5 =2 2+5 =2 7 =128. Example 10. (a – 4c +x) 2 (a – 4c +x) 3 =(a – 4c + x) 5.

4. When dividing powers with the same bases, the exponent of the divisor is subtracted from the exponent of the dividend

Example 11. 12 5:12 3 =12 5-3 =12 2 =144. Example 12. (x-y) 3:(x-y) 2 =x-y.

5. When raising a degree to a power, the exponents are multiplied:

Example 13. (2 3) 2 =2 6 =64. Example 14.

www.maths.yfa1.ru

Powers and roots

Operations with powers and roots. Degree with negative ,

zero and fractional indicator. About expressions that have no meaning.

Operations with degrees.

1. When multiplying powers with the same base, their exponents are added:

a m · a n = a m + n .

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



3. The degree of the product of two or more factors is equal to the product of the degrees of these factors.

4. The degree of a ratio (fraction) is equal to the ratio of the degrees of the dividend (numerator) and divisor (denominator):

(a/b) n = a n / b n .

5. When raising a power to a power, their exponents are multiplied:

All the above formulas are read and executed in both directions from left to right and vice versa.

EXAMPLE (2 3 5 / 15)² = 2² · 3² · 5² / 15² = 900 / 225 = 4 .

Operations with roots. In all the formulas below, the symbol means arithmetic root(the radical expression is positive).

1. The root of the product of several factors is equal to the product of the roots of these factors:

2. The root of a ratio is equal to the ratio of the roots of the dividend and the divisor:



3. When raising a root to a power, it is enough to raise to this power radical number:



4. If you increase the degree of the root by m times and at the same time raise the radical number to the mth power, then the value of the root will not change:



5. If you reduce the degree of the root by m times and simultaneously extract the mth root of the radical number, then the value of the root will not change:





Expanding the concept of degree. So far we have considered degrees only with natural exponents; but operations with powers and roots can also lead to negative, zero various variables fractional indicators. All these exponents require additional definition.

A degree with a negative exponent. The power of a certain number with a negative (integer) exponent is defined as one divided by the power of the same number with an exponent equal to the absolute value of the negative exponent:



Now the formula a m : a n = a m - n can be used not only for m, more than n, but also with m, less than n .

EXAMPLE a 4: a 7 = a 4 - 7 = a - 3 .

If we want the formula a m : a n = a m - n was fair when m = n, we need a definition of degree zero.

A degree with a zero index. The power of any non-zero number with exponent zero is 1.

EXAMPLES. 2 0 = 1, ( – 5) 0 = 1, (– 3 / 5) 0 = 1.

Degree with a fractional exponent. In order to raise a real number a to the power m / n, you need to extract the nth root of the mth power of this number a:



About expressions that have no meaning. There are several such expressions.

Where a ≠ 0 , does not exist.

In fact, if we assume that x is a certain number, then in accordance with the definition of the division operation we have: a = 0· x, i.e. a= 0, which contradicts the condition: a ≠ 0

- any number.

In fact, if we assume that this expression is equal to some number x, then according to the definition of the division operation we have: 0 = 0 · x. But this equality occurs when any number x, which was what needed to be proven.

0 0 - any number.



Solution. Let's consider three main cases:

1) x = 0 – this value does not satisfy this equation

2) when x> 0 we get: x/x= 1, i.e. 1 = 1, which means

What x– any number; but taking into account that in

in our case x> 0, the answer is x > 0 ;

Properties of degree

We remind you that in this lesson we will understand properties of degrees with natural indicators and zero. Powers with rational exponents and their properties will be discussed in lessons for 8th grade.

A power with a natural exponent has several important properties that allow us to simplify calculations in examples with powers.

A power with a natural exponent has several important properties that allow us to simplify calculations in examples with powers.
Property No. 1

When multiplying powers with the same bases, the base remains unchanged, and the exponents of the powers are added.

a m · a n = a m + n, where “a” is any number, and “m”, “n” are any natural numbers.

a m · a n = a m + n, where “a” is any number, and “m”, “n” are any natural numbers.

• Simplify the expression.
  b b 2 b 3 b 4 b 5 = b 1 + 2 + 3 + 4 + 5 = b 15
• Present it as a degree.
  6 15 36 = 6 15 6 2 = 6 15 6 2 = 6 17
• Present it as a degree.
  (0.8) 3 · (0.8) 12 = (0.8) 3 + 12 = (0.8) 15

Please note that in the specified property we were talking only about the multiplication of powers with the same bases. It does not apply to their addition.

You cannot replace the sum (3 3 + 3 2) with 3 5. This is understandable if
calculate (3 3 + 3 2) = (27 + 9) = 36, and 3 5 = 243

calculate (3 3 + 3 2) = (27 + 9) = 36, and 3 5 = 243
Property No. 2

When dividing powers with the same bases, the base remains unchanged, and the exponent of the divisor is subtracted from the exponent of the dividend.

• Write the quotient as a power
  (2b) 5: (2b) 3 = (2b) 5 − 3 = (2b) 2
• Calculate.

11 3 − 2 4 2 − 1 = 11 4 = 44
Example. Solve the equation. We use the property of quotient powers.
3 8: t = 3 4

Answer: t = 3 4 = 81

Answer: t = 3 4 = 81

Example. Simplify the expression.
4 5m + 6 4 m + 2: 4 4m + 3 = 4 5m + 6 + m + 2: 4 4m + 3 = 4 6m + 8 − 4m − 3 = 4 2m + 5

4 5m + 6 4 m + 2: 4 4m + 3 = 4 5m + 6 + m + 2: 4 4m + 3 = 4 6m + 8 − 4m − 3 = 4 2m + 5

2 11 − 5 = 2 6 = 64

Please note that in Property 2 we were only talking about dividing powers with the same bases.

You cannot replace the difference (4 3 −4 2) with 4 1. This is understandable if you calculate (4 3 −4 2) = (64 − 16) = 48, and 4 1 = 4

Be careful!
Property No. 3

When raising a degree to a power, the base of the degree remains unchanged, and the exponents are multiplied.

(a n) m = a n · m, where “a” is any number, and “m”, “n” are any natural numbers.

Example.
(a 4) 6 = a 4 6 = a 24

Example. Express 3 20 as a power with a base of 3 2.

By the property of raising a degree to a power It is known that when raised to a power, exponents are multiplied, which means:

Properties 4
Product power

When a power is raised to a product power, each factor is raised to that power and the results are multiplied.

(a b) n = a n b n, where “a”, “b” are any rational numbers; "n" is any natural number.

• Example 1.
  (6 a 2 b 3 c) 2 = 6 2 a 2 2 b 3 2 c 1 2 = 36 a 4 b 6 c 2
• Example 2.
  (−x 2 y) 6 = ((−1) 6 x 2 6 y 1 6) = x 12 y 6

Please note that property No. 4, like other properties of degrees, is also applied in reverse order.

(a n b n)= (a b) n

That is, to multiply powers with the same exponents, you can multiply the bases, but leave the exponent unchanged.

• Example. Calculate.
  2 4 5 4 = (2 5) 4 = 10 4 = 10,000
• Example. Calculate.
  0.5 16 2 16 = (0.5 2) 16 = 1

In more complex examples, there may be cases where multiplication and division must be performed over powers with different bases and different exponents. In this case, we advise you to do the following.

For example, 4 5 3 2 = 4 3 4 2 3 2 = 4 3 (4 3) 2 = 64 12 2 = 64 144 = 9216

4 5 3 2 = 4 3 4 2 3 2 = 4 3 (4 3) 2 = 64 12 2 = 64 144 = 9216

4 21 (−0.25) 20 = 4 4 20 (−0.25) 20 = 4 (4 (−0.25)) 20 = 4 (−1) 20 = 4 1 = 4

Properties 5
Power of a quotient (fraction)

To raise a quotient to a power, you can raise the dividend and the divisor separately to this power, and divide the first result by the second.

(a: b) n = a n: b n, where “a”, “b” are any rational numbers, b ≠ 0, n - any natural number.

• Example. Present the expression as a quotient of powers.
  (5: 3) 12 = 5 12: 3 12

We remind you that a quotient can be represented as a fraction. Therefore, we will dwell on the topic of raising a fraction to a power in more detail on the next page.

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First, let's remember the basic formulas of powers and their properties.

Product of a number a occurs on itself n times, we can write this expression as a a … a=a n

1. a 0 = 1 (a ≠ 0)

3. a n a m = a n + m

4. (a n) m = a nm

5. a n b n = (ab) n

7. a n / a m = a n - m

Power or exponential equations– these are equations in which the variables are in powers (or exponents), and the base is a number.

Examples of exponential equations:

In this example, the number 6 is the base; it is always at the bottom, and the variable x degree or indicator.

Let us give more examples of exponential equations.
2 x *5=10
16 x - 4 x - 6=0

Now let's look at how exponential equations are solved?

Let's take a simple equation:

2 x = 2 3

This example can be solved even in your head. It can be seen that x=3. After all, in order for the left and right sides to be equal, you need to put the number 3 instead of x.
Now let’s see how to formalize this decision:

2 x = 2 3
x = 3

In order to solve such an equation, we removed identical grounds(that is, twos) and wrote down what was left, these are degrees. We got the answer we were looking for.

Now let's summarize our decision.

Algorithm for solving the exponential equation:
1. Need to check the same whether the equation has bases on the right and left. If the reasons are not the same, we are looking for options to solve this example.
2. After the bases become the same, equate degrees and solve the resulting new equation.

Now let's look at a few examples:

Let's start with something simple.

The bases on the left and right sides are equal to the number 2, which means we can discard the base and equate their degrees.

x+2=4 The simplest equation is obtained.
x=4 - 2
x=2
Answer: x=2

In the following example you can see that the bases are different: 3 and 9.

3 3x - 9 x+8 = 0

First, move the nine to the right side, we get:

Now you need to make the same bases. We know that 9=3 2. Let's use the power formula (a n) m = a nm.

3 3x = (3 2) x+8

We get 9 x+8 =(3 2) x+8 =3 2x+16

3 3x = 3 2x+16 Now it is clear that on the left and right sides the bases are the same and equal to three, which means we can discard them and equate the degrees.

3x=2x+16 we get the simplest equation
3x - 2x=16
x=16
Answer: x=16.

Let's look at the following example:

2 2x+4 - 10 4 x = 2 4

First of all, we look at the bases, bases two and four. And we need them to be the same. We transform the four using the formula (a n) m = a nm.

4 x = (2 2) x = 2 2x

And we also use one formula a n a m = a n + m:

2 2x+4 = 2 2x 2 4

Add to the equation:

2 2x 2 4 - 10 2 2x = 24

We gave an example for the same reasons. But other numbers 10 and 24 bother us. What to do with them? If you look closely you can see that on the left side we have 2 2x repeated, and here is the answer - we can put 2 2x out of brackets:

2 2x (2 4 - 10) = 24

Let's calculate the expression in brackets:

2 4 - 10 = 16 - 10 = 6

We divide the entire equation by 6:

Let's imagine 4=2 2:

2 2x = 2 2 bases are the same, we discard them and equate the degrees.
2x = 2 is the simplest equation. Divide it by 2 and we get
x = 1
Answer: x = 1.

Let's solve the equation:

9 x – 12*3 x +27= 0

Let's convert:
9 x = (3 2) x = 3 2x

We get the equation:
3 2x - 12 3 x +27 = 0

Our bases are the same, equal to three. In this example, you can see that the first three has a degree twice (2x) than the second (just x). In this case, you can solve replacement method. We replace the number with the smallest degree:

Then 3 2x = (3 x) 2 = t 2

We replace all x powers in the equation with t:

t 2 - 12t+27 = 0
We get a quadratic equation. Solving through the discriminant, we get:
D=144-108=36
t 1 = 9
t2 = 3

Returning to the variable x.

Take t 1:
t 1 = 9 = 3 x

That is,

3 x = 9
3 x = 3 2
x 1 = 2

One root was found. We are looking for the second one from t 2:
t 2 = 3 = 3 x
3 x = 3 1
x 2 = 1
Answer: x 1 = 2; x 2 = 1.

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