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. \({625^{\frac{1}{4}}}\)
Answer:
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. \({\left( {\frac{{27}}{{64}}} \right)^{\frac{2}{3}}}\)
Answer:
a.
16 3 2 = ( 16 ) 3 = 4 3 =64
b.
( 27 64 ) 2 3 = ( 27 64 3 ) 2 = ( 3 4 ) 2 = 9 16