Fractional Indices

Fractional Indices

$${a^{\frac{1}{n}}}$$ is a nth root of a. ${a^{\frac{1}{n}}} = \sqrt[n]{a}$ $${a^{\frac{m}{n}}}$$ means the nth root of $${a^m}$$.
$a m n = a m n$
Example:
Find the value of the following:
a. $${81^{\frac{1}{2}}}$$
b. $${64^{\frac{1}{3}}}$$
c. $${625^{\frac{1}{4}}}$$
a.
${81^{\frac{1}{2}}} = \sqrt {81} = 9$
b.
${64^{\frac{1}{3}}} = \sqrt[3]{{64}} = 4$
c.
${625^{\frac{1}{4}}} = \sqrt[4]{{625}} = 5$

Example:
Find the value of the following:
a. $${16^{\frac{3}{2}}}$$
b. $${\left( {\frac{{27}}{{64}}} \right)^{\frac{2}{3}}}$$
$16 3 2 = ( 16 ) 3 = 4 3 =64$
$( 27 64 ) 2 3 = ( 27 64 3 ) 2 = ( 3 4 ) 2 = 9 16$