Simplify

Solution:

(i) $\left(\frac{15^{\frac{1}{3}}}{9^{\frac{1}{4}}}\right)^{-6}$

$\left(\frac{15^{\frac{1}{3}}}{9^{\frac{1}{4}}}\right)^{-6}=\left(\frac{9^{\frac{1}{4}}}{15^{\frac{1}{3}}}\right)^{6}$

$=\frac{9 \frac{6}{4}}{15^{\frac{6}{3}}}$

$=\frac{9^{\frac{3}{2}}}{15^{2}}$

$=\frac{\left(3^{2}\right)^{\frac{3}{2}}}{15^{2}}$

$=\frac{3^{3}}{15^{2}}$

$=\frac{27}{225}$

$=\frac{3}{25}$

(ii) $\left(\frac{12^{\frac{1}{5}}}{27^{\frac{1}{5}}}\right)^{\frac{5}{2}}$

$\left(\frac{12^{\frac{1}{5}}}{27^{\frac{1}{5}}}\right)^{\frac{5}{2}}=\frac{\left(12^{\frac{1}{5}}\right)^{\frac{5}{2}}}{\left(27^{\frac{1}{5}}\right)^{\frac{5}{2}}}$

$=\frac{12 \frac{1}{2}}{27^{\frac{1}{2}}}$

$=\left(\frac{12}{27}\right)^{\frac{1}{2}}$

$=\left(\frac{4}{9}\right)^{\frac{1}{2}}$

$=\left[\left(\frac{2}{3}\right)^{2}\right]^{\frac{1}{2}}$

$=\frac{2}{3}$

(iii) $\left(\frac{15^{\frac{1}{4}}}{3^{\frac{1}{2}}}\right)^{-2}$

$\left(\frac{15^{\frac{1}{4}}}{3^{\frac{1}{2}}}\right)^{-2}=\left(\frac{3^{\frac{1}{2}}}{15^{\frac{1}{4}}}\right)^{2}$

$=\frac{3^{\frac{2}{2}}}{15^{\frac{2}{4}}}$

$=\frac{3}{15^{\frac{1}{2}}}$

$=\frac{3}{3^{\frac{1}{2}} .5^{\frac{1}{2}}}$

$=\frac{3^{1-\frac{1}{2}}}{5^{\frac{1}{2}}}$

$=\frac{3^{\frac{1}{2}}}{5^{\frac{1}{2}}}$

$=\sqrt{\frac{3}{5}}$