Question

Write the following numbers in the form of rational indices.
(i) Cube root of 8th power of 548
(ii) Square of 3rd root of 131

Answer 1

It is known that,
$(a{)}^{\frac{m}{n}}=(a^m{)}^{\frac{1}{n}}{\text{mean}}^{\prime }{n}^{th}\text{root of}{m}^{th}\text{power of a'.}$
$(a{)}^{\frac{m}{n}}=(a^{\frac{1}{n}}{)}^m{\text{means}}^{\prime }{m}^{th}\text{power of}{n}^{th}\text{root of a'.}$

(i) Cube root of $8^{th}$ power of 548
$=\left\{{\left({548}^8\right)}^{\frac{1}{3}}\right\}=(548{)}^{\frac{8}{3}}$
(ii) Square of 3 ${}^{rd}$ root of 131
$=\left\{{\left({131}^{\left(\frac{1}{3}\right)}\right)}^2\right\}=(131{)}^{\frac{2}{3}}$