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Law of Fractional Indices

Fractional indices are expressions in surds or irrational form. In other words, the expressions have index or indices in fraction form.

To understand fractional indices, consider the following examples:

Example 1:

62  =  36    Therefore sqrt{36}  =  6

23  =  8    Therefore root{3}{8}  =  2

Now consider 16^{1/2}~*~16^{1/2}  =  16^({1/2}~+~{1/2})  =  16

16^{1/2} multiplied by itself gives 16, and sqrt{16} multiplied by itself gives 16.

Hence 16^{1/2} and sqrt{16} are the same.

16^{1/2}  =  sqrt{16}

Summarising the above, we can conclude that x^{1/2}  =  sqrt{x}

 

Example 2: Let’s now look at 64^{1/3}~*~64^{1/3}~*~64^{1/3}

=  64^({1/3}~+~{1/3}~+~{1/3})  =  64

We can express the above expression also as root{3}{64}~*~root{3}{64}~*~root{3}{64}  =  64

So 64^{1/3} multiplied by itself thrice gives 64, and root{3}{64} multiplied by itself thrice also gives 64.

Hence 64^{1/3}  =  root{3}{64}

So we can conclude that x^{1/3}  =  root{3}{x} meaning x power of one-third is equal to the cube root of x.

This leads to the generalisation that x^{1/n}  =  root{n}{x} meaning x power of one-n is equal to the nth root of x.

 

Learn about law of negative indices here