# EVALUATE EXPRESSIONS USING PROPERTIES OF EXPONENTS

## About "Evaluate expressions using properties of exponents"

Evaluate expressions using properties of exponents :

To evaluate expressions, we use the following properties.

Product rule :

If we have same base for two or more terms which are multiplying , we have to write only one base and add the powers

am x an  =  a(m+n)

Quotient rule :

If we have same base for two or more terms which are dividing, we have to write only one base and subtract the powers

am / an  =  a(m-n)

Power rule :

When we have power raised to another power, we have to multiply both powers.

(am)n  =  amn

Reciprocal rule :

If we want to change the negative power as positive, we have to take its reciprocal of the base and change the sign.

a-m =  1/am

(a/b)-m  =  (b/a)m

Numbers with zero exponent :

The value of any thing to power zero.

a0 =  1

Multiplying numbers with same exponents :

In general, for any two integers a and b we have

a2 b 2 = (a x b)2  =  (ab)2

Let us look into some examples to understand the above concept.

## Evaluate expressions using properties of exponents - Examples

Example 1 :

Find the value of the following

34 x 3-3

Solution :

=  34 x 3-3

Since we have same base for both terms which are multiplying, we need to write only one base and combine the powers.

=  3(4 - 3)

=  31  ==>  3

Example 2 :

Find the value of the following

1/3-4

Solution :

=  1/3-4

In order to convert the negative power as positive, we need to take the reciprocal.

1/3-4  =  34

=  3 x 3 x 3 x 3

=  81

Example 3 :

Find the value of the following

(4/5)2

Solution :

=  (4/5)2

Since we have common power for both numerator and denominator, we need to distribute the power.

(4/5)=  42/52

=  (4 x 4)/(5 x 5)

=  16/25

Example 4 :

Find the value of the following

10-3

Solution :

=  10-3

In order to convert the negative power as positive, we need to take its reciprocal.

10-3  =  1/103

=  1/(10 x 10 x 10)

=  1/1000

Example 5 :

Find the value of the following

(-1/2)5

Solution :

=  (-1/2)5

Since we have common power for the fraction, we need to distribute the power for both numerator and denominator.

(-1/2) =  (-1)5/25

The value of (-1)5 is -1

The value of 25 is 2 x 2 x 2 x 2 x 2 = 32

(-1)5/2 =  -1/32

Example 6 :

Find the value of the following

(7/4)0 x 3

Solution :

=  (7/4)x 3

The value of anything to the power zero is 1.

=  1 x 3  ==>  3

Example 7 :

Find the value of the following

(23)-2 x (32)2

Solution :

=  (23)-2 x (32)2

When we have power raised to another power, we have to multiply both powers.

=  2-6 x 34

In order to convert the negative power as positive, we need to take its reciprocal.

=  (1/26) x 34

=  (1/64) x 81

=  81/64

Example 8 :

Find the value of the following

(3/8)x (3/8)4÷ (3/8)9

Solution :

=  (3/8)x (3/8)÷ (3/8)9

Let us distribute the power for numerator and denominators.

=  (35/85) x (34/84÷ (3/8)9

=  3(5 + 4)/8(5+4) ÷ (3/8)9

=  39/89 ÷ (3/8)9

=  1

Example 9 :

Find the value of the following

(-2)x (-2)-6

Solution :

=  (-2)x (-2)-6

=  (-2)(5 - 6)

=  (-2)- 1

=  1/(-2) 1

=  -1/2

Example 10 :

Find the value of the following

(20 + 4-1) x (-2)6

Solution :

=  (2+ 4-1) x (-2)6

=  (1 +(1/4)) x (-2)6

=  (4 +1)/4 x (-2)6

=  5/4 x 64

=  5 x 16

=  80

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