# EXPONENT PROPERTIES INVOLVING PRODUCTS

In the page "Exponent properties involving products" we are going to learn about how to multiply two or more terms involving exponents.

## What is exponent?

The exponent of a number says how many times to use the number in a multiplication.

for example 5³ = 5 x 5 x 5

In words 5³ could be called as 5 to the power 3 or 5 cube.

## Basic Exponent Rules:

Rule 1:

When we have to simplify two or more the terms which are multiplying with same base,then we have to put the same base and add the powers.

Rule 2:

Whenever we have two terms which are diving with the same base,we have to put only one base and we have to subtract the powers.

Rule 3:

Whenever we have power to the power,we have to multiply both powers.

Rule 4:

Anything to the power zero is 1.

Rule 5:

If we have same power for 2 or more terms which are multiplying or dividing,we have to apply the powers for every terms.

Note:

This rule is not applicable when two are more terms which are adding and subtracting.

For example (x + y) ^m = (x^m + y^m) is not correct

## How to move an exponents or powers to the other side ?

 If the power goes from one side of equal sign to the other side,it will flip.that is x = 4²

## What is exponent and power?

The other names of exponent are index and power.

Other things:

Point 1:

If we don't have any number in the power then we have to consider that there is 1

## Negative numbers with fractions

Point 2:

Incase we have negative power for any fraction and we want to make it as positive, we can write the power as positive and we should write its reciprocal only. For example

## Negative numbers with negative exponents

Find the value of (-5) ³

To find value of the above expression, first we have to consider the power.

• If the power is odd, then the answer will have negative sign.
• If the power is even, then the answer will have positive sign.

here, the power of -5 is 3, that is odd. So, the answer will have negative sign and we have to multiply the base three times.

Finally we will get -125 as the answer.

Procedure of multiplying two polynomials with exponents

We will multiply two or more polynomials in the following order.

(1) Symbol

(2) Number

(3) Variable

Let us see how it works

Multiply ( 5 x² ) and (-2 x³)

=  ( 5 x² )  x (-2 x³)

=  10 x

## Example problems of exponent properties involving products

Question 1 :

Simplify (5x²)⁴ x (2x)³

Solution :

=  (5x²)⁴ x (2x)³

5⁴ = 5 x 5 x 5 x 5

2³ = 2 x 2 x 2

Since x⁸ and x³ are having same base, we can combine these terms.

Let us see the next example on "Exponent properties involving products"

Question 2 :

If 5 √5 x 5³= 5 ²⁾ then find the value of n.

Solution :

Let us see the next example on "Exponent properties involving products"

Question 3:

Simplify  (52 x⁶/13x⁷)

Solution :

=  (52 x⁶/13x⁷)

We have negative power for the x term which is in the denominator.To make it as positive we are going to write it in numerator.

=  (52 xx/13)

=  4 x

=  4 x¹³

Let us see the next example on "Exponent properties involving products"

Question 4 :

Simplify  (-7x²) (x⁴)

Solution :

=  (-7x²) (x⁴)

=  -7 x²

=  -7 x

Question 5 :

Simplify  (10 x y³ z²) (-2 x y⁵z)

Solution :

=  (10 x y³ z²) (-2 x y⁵z)

=  -20 x^(1+1) y^(3+5) z^(2+1)

= -20 x²y⁸z³

Question 6 :

Simplify  (-5 xyz) (4 y⁵z) y³

Solution :

=   (-5 xyz) (4 y⁵z) y³

-20 x yy³z

= -20 x y^(5+3) z^(1+1)

= - 20 x yz²

Question 7 :

Simplify  (7 a b² c³) (¾ a b c⁴)

Solution :

=   (7 a b² c³) (¾ a b c⁴)

(21/4) a^(1+1)b^(2+1)c^(3+4)

= (21/4) a²b³c

Question 8 :

Simplify  (-4 a³) (-5 a⁴)

Solution :

=    (-4 a³) (-5 a⁴)

20 a^(3+4)

= 20 a

Question 9 :

Simplify  (2 a³) (¼) a

Solution :

=    (2 a³) (¼) a

(2/4)a^(3+4)

= (1/2) a

Question 10 :

Simplify  (10 p³) (7/25) p

Solution :

=   (10 p³) (7/25) p

10(7/25)p^(3+7)

= (1/2) a¹

After having gone through the stuff given above, we hope that the students would have understood "Exponent properties involving products".

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