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Fractional Exponent Law. How to work out the value of fraction exponents.

Updated on September 13, 2011


A fractional exponent can be worked out by applying the following rule:

am/n = (n√a)m

This means the nth root of a raised to the power of m. This is easier to see if you take a look at some numerical examples:

Example 1

Work out the value of 163/2.

All you need to do is apply the above rule for working out negative exponents:

163/2 = (2√16)3

So first work out what’s inside the bracket, that is, the square root of 16 (2√16 is the same as √16):

(2√16)3 = 43 (since the square root of 16 is 4)

Next work out the cube of 4 to give the final answer:

43 = 64 (since 4 × 4 × 4 = 64)

Example 2

Work out the value of 811/2.

Again, all you need to do is apply the above rule for working out negative exponents:

811/2 = (2√81)1

So first work out what’s inside the bracket, that is, the square root of 81 (2√81 is the same as √81):

(2√81)1 = 91 (since the square root of 81 is 9)

Next work out the 9 to the power of 1 to give the final answer:

91 = 9

Example 3

Work out the value of 82/3.

Again, all you need to do is apply the above rule for working out negative exponents:

82/3 = (3√8)2

So first work out what’s inside the bracket, that is, the cube root of 8:

(3√8)2 = 22 (since the cube root of 8 is 2)

Next work out the square of 2:

22 = 4 (since 2× 2 = 4)

Example 4

Work out the value of (64/125)1/3.

First of all (64/125)1/3 can be written as 641/3/1251/3

Next, apply the negative exponent rule separately to the numerator and denominator of the fraction you have just made:

641/3= (3√64)1

1251/3 = (3√125)1

So all you need to do now is work out the cube root of both numbers:

(3√64)1 = 4

(3√125)1 = 5

So the final answer to (64/125)1/3 = 4/5

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