# EVALUATE RATIONAL EXPONENTS

Evaluate rational exponents :

In this topic we will have questions with fractional or rational exponent.

Before learning evaluate rational exponents, first let us think about rules of exponents.

(i) x^m x x^n = x ^(m + n)

(ii) x^m / x^n = x ^(m - n)

(iii) (x^m)^n = x ^(mn)

(iv) x^(-m)= 1/x ^m

We have to follow the above rules while evaluating rational exponents.

Let us see discuss the steps in detail in the example problem given below.

## Evaluate rational exponents - Examples

Example 1 :

Evaluate 100^(1/2)

Solution :

=  100^(1/2)

=  (10 x 10)^(1/2)

Instead of writing 10 two times we can write 10².

=  (10^2)^(1/2) ==> 10^[2 x (1/2)]

=  10^1 ==>  10

Hence, the value of 100^(1/2) is 10.

Example 2 :

Evaluate 16^(1/4)

Solution :

=  16^(1/4)

=  (2 x 2 x 2 x 2)^(1/4)

Instead of writing 2 four times we can write 2.

=  (2^4)^(1/4) ==> 2^[4 x (1/4)]

=  2^1 ==>  2

Hence, the value of 16^(1/4) is 2.

Example 3 :

Evaluate 25^(3/2)

Solution :

=  25^(3/2)

=  (5 x 5)^(3/2)

Instead of writing 5 two times we can write 5².

=  (5^2)^(3/2) ==> 5^[2 x (3/2)]

=  5^3 ==>  5 x 5 x 5 = 125

Hence, the value of 25^(3/2) is 125.

Example 4 :

Evaluate 121^(-1/2)

Solution :

=  121^(-1/2)

To make the negative power as positive, we have to flip the base.

=  1/[121^(1/2)]

121 can be written as 11 x 11 = 11²

=  1/[(11^2)(1/2)]

=  1/[11^(2 x (1/2))]

= 1/11

Hence, the value of 121^(-1/2) is 1/11.

Example 5 :

Evaluate 5^3/2 x 5^1/4

Solution :

=   5^3/2 x 5^1/4

Since we have same base, we have to write one base and add the powers.

=   5^[(3/2) + (1/4 )]

=   5^[(6/4) + (1/4 )]

=   5^[(6+1)/4]

=   5^7/4

Example 6 :

Find the value of x, when x^3  =  8

Solution :

x^3  =  8

Now we are going to write as the multiple of 2.

8 = 2 x 2 x 2 = 2^3

Since we have same base, we have to write one base and add the powers.

=   5^[(3/2) + (1/4 )]

=   5^[(6/4) + (1/4 )]

=   5^[(6+1)/4]

=   5^7/4

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