If the indicators are the same but the bases are different. Multiplication and division of numbers with powers

21.10.2019

In the last video tutorial, we learned that the degree of a certain base is an expression that is the product of the base and itself, taken in an amount equal to the exponent. Let's now explore some the most important properties and operations of powers.

For example, let's multiply two different powers with the same base:

Let's take a look at this piece in its entirety:

(2) 3 * (2) 2 = (2)*(2)*(2)*(2)*(2) = 32

Calculating the value of this expression, we get the number 32. On the other hand, as can be seen from the same example, 32 can be represented as a product of the same base (two), taken 5 times. And indeed, if you count, then:

Thus, it can be safely concluded that:

(2) 3 * (2) 2 = (2) 5

This rule works successfully for any indicators and any grounds. This property of multiplication of the degree follows from the rule of preservation of the meaning of expressions during transformations in the product. For any base a, the product of two expressions (a) x and (a) y is equal to a (x + y). In other words, when producing any expressions with the same base, the final monomial has a total degree formed by adding the degree of the first and second expressions.

The presented rule also works great when multiplying several expressions. The main condition is that the bases for all be the same. For instance:

(2) 1 * (2) 3 * (2) 4 = (2) 8

It is impossible to add degrees, and in general to carry out any power joint actions with two elements of the expression, if their bases are different.
As our video shows, due to the similarity of the processes of multiplication and division, the rules for adding powers during a product are perfectly transferred to the division procedure. Consider this example:

Let's perform a term-by-term transformation of the expression into full view and cancel the same elements in the dividend and divisor:

(2)*(2)*(2)*(2)*(2)*(2) / (2)*(2)*(2)*(2) = (2)(2) = (2) 2 = 4

The end result of this example is not so interesting, because already in the course of its solution it is clear that the value of the expression is equal to the square of two. And it is the deuce that is obtained by subtracting the degree of the second expression from the degree of the first.

To determine the degree of the quotient, it is necessary to subtract the degree of the divisor from the degree of the dividend. The rule works with the same basis for all its values and for all natural powers. In abstract form, we have:

(a) x / (a) y = (a) x - y

The definition for the zero degree follows from the rule for dividing identical bases with powers. Obviously, the following expression is:

(a) x / (a) x \u003d (a) (x - x) \u003d (a) 0

On the other hand, if we divide in a more visual way, we get:

(a) 2 / (a) 2 = (a) (a) / (a) (a) = 1

When reducing all visible elements of a fraction, the expression 1/1 is always obtained, that is, one. Therefore, it is generally accepted that any base raised to the zero power is equal to one:

Regardless of the value of a.

However, it would be absurd if 0 (which still gives 0 for any multiplication) is somehow equal to one, so an expression like (0) 0 (zero to the zero degree) simply does not make sense, and to formula (a) 0 = 1 add a condition: "if a is not equal to 0".

Let's do the exercise. Let's find the value of the expression:

(34) 7 * (34) 4 / (34) 11

Since the base is the same everywhere and equals 34, the final value will have the same base with a degree (according to the above rules):

In other words:

(34) 7 * (34) 4 / (34) 11 = (34) 0 = 1

Answer: The expression is equal to one.

Degree with a negative exponent. Division of powers with the same base. 4. Reduce the exponents 2a4/5a3 and 2/a4 and bring them to a common denominator. The base and argument of the first logarithm are exact powers. This property extends to the power of the product of three or more factors. Therefore, am−an>0 and am>an, which was to be proved. It remains to prove the last of the listed properties of powers with natural exponents.

Please note that property No. 4, like other properties of degrees, is also applied in reverse order. That is, to multiply degrees with the same exponents, you can multiply the bases, and leave the exponent unchanged. The calculation of the power value is called the exponentiation action. That is, when calculating the value of an expression that does not contain brackets, first perform the action of the third step, then the second (multiplication and division), and finally the first (addition and subtraction).

After the degree of a number is determined, it is logical to talk about the properties of the degree. In this article, we will give the basic properties of the degree of a number, while touching on all possible exponents. Here we will give proofs of all properties of the degree, and also show how these properties are applied when solving examples. We note right away that all the written equalities are identical under the specified conditions, and their right and left parts can be interchanged.

Let us give an example that confirms the main property of the degree. Before giving the proof of this property, let us discuss the meaning of the additional conditions in the formulation. The condition m>n is introduced so that we do not go beyond natural exponents. The main property of a fraction allows us to write the equality am−n·an=a(m−n)+n=am.

Transition to a new foundation

That is, the natural degree property n of the product of k factors is written as (a1·a2·…·ak)n=a1n·a2n·…·akn. For clarity, we show this property with an example. The proof can be carried out using the previous property. For example, for any natural numbers p, q, r and s are equal. For greater clarity, let's give an example with specific numbers: (((5,2)3)2)5=(5,2)3+2+5=(5,2)10.

This fact and the properties of multiplication allow us to assert that the result of multiplying any number of positive numbers will also be a positive number. It is quite obvious that for any natural n with a=0 the degree of an is zero. Indeed, 0n=0·0·…·0=0. For example, 03=0 and 0762=0. Let's move on to negative bases. Let's start with the case when the exponent is an even number, denote it as 2·m, where m is a natural number.

We turn to the proof of this property. Let us prove that for m>n and 0 By the same principle it is possible to prove all other properties of the degree with an integer exponent, written as equalities. Conditions p 0 in this case will be equivalent to conditions m 0, respectively. In this case, the condition p>q will correspond to the condition m1>m2, which follows from the comparison rule ordinary fractions With same denominators.

Operations with roots. Extension of the concept of degree. So far, we have considered exponents with natural exponents only; but actions with exponents and roots can also lead to negative, zero, and fractional exponents. All these exponents require an additional definition. If we want the formula a m: a n=a m - n to be valid for m = n, we need to define the zero degree. Logarithms, like any number, can be added, subtracted and converted in every possible way.

Removing the exponent from the logarithm

If the bases are different, these rules do not work! Speaking about the rules for adding and subtracting logarithms, I specifically emphasized that they only work with the same bases. It follows from the second formula that it is possible to interchange the base and the argument of the logarithm, but in this case the whole expression is “turned over”, i.e. the logarithm is in the denominator.

It is possible to evaluate how convenient they are only when deciding logarithmic equations and inequalities. Since the product does not change from permutation of factors, we calmly multiplied four and two, and then figured out the logarithms. Often in the process of solving it is required to represent a number as a logarithm to a given base.

Properties of degrees, formulations, proofs, examples.

The number n can be absolutely anything, because it's just the value of the logarithm. That's what it's called: basic logarithmic identity. Like the new base conversion formulas, the basic logarithmic identity is sometimes the only possible solution. In conclusion, I will give two identities that are difficult to call properties - rather, these are consequences from the definition of the logarithm.

Examples of solving examples with fractions containing numbers with powers

Remember once and for all: the logarithm to any base a from that base itself is equal to one. 1 = 0 is logarithmic zero. The base a can be anything, but if the argument is one - the logarithm is zero! Because a0 = 1 is a direct consequence of the definition. That's all the properties. Download the cheat sheet at the beginning of the lesson, print it out - and solve the problems.

Logarithmic unit and logarithmic zero

2.a-4 is a-2 the first numerator. In this case, we advise you to do the following. This is the third stage action. For example, the main property of the fraction am·an=am+n, when simplifying expressions, is often used in the form am+n=am·an. The condition a≠0 is necessary in order to avoid division by zero, since 0n=0, and when we got acquainted with division, we agreed that it is impossible to divide by zero. From the resulting equality am−n·an=am and from the connection between multiplication and division, it follows that am−n is a quotient of am and an. This proves the property of partial powers with the same bases.

Similarly, if q=0, then (ap)0=1 and ap 0=a0=1, whence (ap)0=ap 0. In more complex examples, there may be cases when multiplication and division must be performed on powers with different bases and different exponents. These inequalities in the properties of the roots can be rewritten as and respectively. And the definition of a degree with a rational exponent allows us to pass to the inequalities and, respectively.

Division of powers with the same base. The main property of a degree based on the properties of multiplication can be generalized to the product of three or more degrees with the same bases and natural exponents.

3.a-3 is a0 = 1, the second numerator. In more complex examples, there may be cases when multiplication and division must be performed on powers with different bases and different exponents. Now let's take a look at them concrete examples and try to prove it.

Thus, we proved that when dividing two powers with the same bases, their indicators must be subtracted. After the degree of a number is determined, it is logical to talk about the properties of the degree.

Here we will give proofs of all properties of the degree, and also show how these properties are applied when solving examples. For example, the main property of the fraction am·an=am+n, when simplifying expressions, is often used in the form am+n=am·an. Let us give an example that confirms the main property of the degree. Before giving the proof of this property, let us discuss the meaning of the additional conditions in the formulation.

Properties of degrees with natural indicators

The condition m>n is introduced so that we do not go beyond natural exponents. From the resulting equality am−n·an=am and from the connection between multiplication and division, it follows that am−n is a quotient of am and an. This proves the property of partial powers with the same bases. For clarity, we show this property with an example. For example, equality holds for any natural numbers p, q, r, and s. For greater clarity, let's give an example with specific numbers: (((5,2)3)2)5=(5,2)3+2+5=(5,2)10.

Addition and subtraction of monomials

We turn to the proof of this property. Let us prove that for m>n and 0 It remains to prove the second part of the property. Therefore, am−an>0 and am>an, which was to be proved. It is not difficult to prove each of these properties, for this it is enough to use the definitions of the degree with a natural and integer exponent, as well as the properties of actions with real numbers.

If p=0, then we have (a0)q=1q=1 and a0 q=a0=1, whence (a0)q=a0 q. By the same principle, one can prove all other properties of a degree with an integer exponent, written in the form of equalities. Conditions p 0 in this case will be equivalent to conditions m 0, respectively.

In this case, the condition p>q will correspond to the condition m1>m2, which follows from the rule for comparing ordinary fractions with the same denominators. These inequalities in the properties of the roots can be rewritten as and respectively. And the definition of a degree with a rational exponent allows us to pass to the inequalities and, respectively.

Basic properties of logarithms

The calculation of the power value is called the exponentiation action. That is, when calculating the value of an expression that does not contain brackets, first perform the action of the third step, then the second (multiplication and division), and finally the first (addition and subtraction). Operations with roots.

Extension of the concept of degree. So far, we have considered exponents with natural exponents only; but actions with exponents and roots can also lead to negative, zero, and fractional exponents. All these exponents require an additional definition. If we want the formula a m: a n=a m - n to be valid for m = n, we need to define the zero degree.

Multiplying powers of numbers with the same exponents. Next, we formulate a theorem on division of powers with equal bases, solve explanatory problems, and prove the theorem in the general case. We now turn to the definition of negative powers. You can easily verify this by substituting the formula from the definition into the rest of the properties. To solve this problem, remember that: 49 = 7^2 and 147 = 7^2 * 3^1. If you now carefully use the properties of degrees (when raising a degree to a power, exponents ...

That is, the exponents are indeed subtracted, but since the exponent is negative in the denominator of the exponent, when subtracting minus by minus, it gives a plus, and the exponents are added. Let's remember what is called a monomial, and what operations can be done with monomials. Recall that in order to reduce a monomial to standard view you must first obtain a numerical coefficient by multiplying all the numerical factors, and then multiply the corresponding powers.

Transition to a new foundation

That is, we must learn to distinguish between similar and non-similar monomials. We conclude: similar monomials have the same letter part, and such monomials can be added and subtracted.

Thank you for your feedback. If you like our project and you are ready to help or take part in it, send information about the project to your friends and colleagues. In the previous video it was said that in examples with monomials there can only be multiplication: “Let's find the difference between these expressions and the previous ones.

The very concept of a monomial as a mathematical unit implies only the multiplication of numbers and variables, if there are other operations, the expression will no longer be a monomial. But at the same time, monomials can be added, subtracted, divided among themselves ... Logarithms, like any numbers, can be added, subtracted and transformed in every possible way. But since logarithms are not quite ordinary numbers, there are rules here, which are called basic properties.

Note: key moment here are the same bases. If the bases are different, these rules do not work! Speaking about the rules for adding and subtracting logarithms, I specifically emphasized that they only work with the same bases. It follows from the second formula that it is possible to interchange the base and the argument of the logarithm, but in this case the whole expression is “turned over”, i.e. the logarithm is in the denominator.

That is, the natural degree property n of the product of k factors is written as (a1·a2·…·ak)n=a1n·a2n·…·akn. There are no rules for adding and subtracting powers with the same base. The base and argument of the first logarithm are exact powers. 4. Reduce the exponents 2a4/5a3 and 2/a4 and bring them to a common denominator.

Obviously, numbers with powers can be added like other quantities , by adding them one by one 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 the same powers of the same variables can be added or subtracted.

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

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

But degrees various variables and various degrees identical variables, must be added by adding them to 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 neither twice the square of a, but 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 subtrahend 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

Power multiplication

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

So, 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 the same 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 sum 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 is;

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

So, powers with the same bases can be multiplied by adding the exponents.

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 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 degree.

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 powers

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

So a 3 b 2 divided by b 2 is 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 powers 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:
y2m: ym = ym
8a n+m: 4a m = 2a n
12(b + y) n: 3(b + y) 3 = 4(b + y) n-3

The rule is also valid for numbers with negative degree values.
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 the 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 in $\frac(5a^4)(3a^2)$ Answer: $\frac(5a^2)(3)$.

2. Reduce the exponents in $\frac(6x^6)(3x^5)$. 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 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 for multiplying and dividing powers with the same and different exponents. Examples"

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

The purpose of the lesson: learn how to perform operations with powers of a number.

To begin with, let's recall the concept of "power of a number". An expression like $\underbrace( a * a * \ldots * a )_(n)$ can be represented as $a^n$.

The reverse 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 base of the degree.
n- exponent.
If n=1, which means the number a taken once and respectively: $a^n= 1$.
If n=0, then $a^0= 1$.

Why this happens, we can find out when we get acquainted with the rules for multiplying and dividing powers.

multiplication rules

a) If powers with the same base are multiplied.
To $a^n * a^m$, we write the powers 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 large power.
Example.
$2^7= 2^3 * 2^4 = 8 * 16 = 128$.

b) If powers are multiplied with a different base, but the same exponent.
To $a^n * b^n$, we write the powers 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 base of the degree is the same, the exponents are different.
Consider dividing a degree with a larger exponent by dividing a degree with a smaller exponent.

So, it is necessary $\frac(a^n)(a^m)$, where n>m.

We 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 a power of zero. 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 you need $\frac(a^n)( b^n)$. We write the powers of numbers as a fraction:

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

Let's imagine for convenience.

Using the property of fractions, we divide a large fraction into a 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.
$\frac(4^3)( 2^3)= (\frac(4)(2))^3=2^3=8$.