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

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

**a ^{m/n} = (^{n}√a)^{m}**

This means the **n**th 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 16^{3/2}.

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

16^{3/2} = (^{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} = 4^{3} (since the square root of 16 is 4)

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

4^{3} = 64 (since 4 × 4 × 4 = 64)

**Example 2**

Work out the value of 81^{1/2}.

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

81^{1/2} = (^{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} = 9^{1} (since the square root of 81 is 9)

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

9^{1} = 9

**Example 3**

Work out the value of 8^{2/3}.

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

8^{2/3} = (^{3}√8)^{2}

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

(^{3}√8)^{2} = 2^{2} (since the cube root of 8 is 2)

Next work out the square of 2:

2^{2} = 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 64^{1/3}/125^{1/3}

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

64^{1/3}= (^{3}√64)^{1}

125^{1/3} = (^{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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