Evaluate
Question:

Evaluate

(i) $(125)^{\frac{1}{3}}$

(ii) $(64)^{\frac{1}{6}}$

(iii) $(25)^{\frac{3}{2}}$

(iv) $(81)^{\frac{3}{4}}$

(v) $(64)^{-\frac{1}{2}}$

(vi) $(8)^{-\frac{1}{3}}$

Solution:

$(\mathrm{i})(125)^{\frac{1}{3}}=\left(5^{3}\right)^{\frac{1}{3}}=5^{3 \times \frac{1}{3}}=5$

$(\mathrm{ii})(64)^{\frac{1}{6}}=\left(2^{6}\right)^{\frac{1}{6}}=2^{6 \times \frac{1}{6}}=2$

$($ iii $)(25)^{\frac{3}{2}}=5^{2 \times \frac{3}{2}}=5^{3}=125 \quad\left[\left(a^{m}\right)^{n}=a^{m n}\right]$

$($ iv $)(81)^{\frac{3}{4}}=\left(3^{4}\right)^{\frac{3}{4}}=3^{4 \times \frac{3}{4}}=3^{3}=27$

$(\mathrm{v})(64)^{\frac{-1}{2}}=\left(8^{2}\right)^{\frac{-1}{2}}=8^{2 \times \frac{-1}{2}}=8^{-1}=\frac{1}{8}$

$(\mathrm{vi})(8)^{\frac{-1}{3}}=\left(2^{3}\right)^{\frac{-1}{3}}=2^{3 \times \frac{-1}{3}}=2^{-1}=\frac{1}{2}$