Question

Write the following numbers in the form of rational indices.
(1) Square root of 5th power of 121
(2) Cube of 4th root of 324
(3) 5th root of square of 264
(4) Cube of cube root of 3

Answer 1

It is known that,

a^m/n = (a^m)^1/n means 'n^th root of m^th power of a'.

a^m/n = (a^1/n)^m means 'm^th power of n^th root of a'.

(1) Square root of 5^th power of 121
$$\begin{gathered} ={\left\{{\left(121\right)}^5\right\}}^{\frac{1}{2}} \\ ={\left(121\right)}^{\frac{5}{2}} \end{gathered}$$
(2) Cube of 4^th root of 324
$$\begin{gathered} ={\left\{{\left(324\right)}^{\frac{1}{4}}\right\}}^3 \\ ={\left(324\right)}^{\frac{3}{4}} \end{gathered}$$
(3) 5^th root of square of 264
$$\begin{gathered} ={\left\{{\left(264\right)}^2\right\}}^{\frac{1}{5}} \\ ={\left(264\right)}^{\frac{2}{5}} \end{gathered}$$
(4) Cube of cube root of 3
$$\begin{gathered} ={\left\{3^{\frac{1}{3}}\right\}}^3 \\ =3^{\frac{3}{3}} \end{gathered}$$