Question:
Simplify:
(i) $\frac{6^{1 / 4}}{6^{1 / 5}}$
(ii) $\frac{8^{1 / 2}}{8^{2 / 3}}$
(iii) $\frac{5^{6 / 7}}{5^{2 / 3}}$
Solution:
$(\mathrm{i}) \frac{6^{\frac{1}{4}}}{6^{\frac{1}{5}}}=6^{\frac{1}{4}-\frac{1}{5}}=6^{\frac{5-4}{20}}=6^{\frac{1}{20}} \quad\left(\frac{a^{m}}{a^{n}}=a^{m-n}\right)$
$(\mathrm{ii}) \frac{8^{\frac{1}{2}}}{8^{\frac{2}{3}}}=8^{\frac{1}{2}-\frac{2}{3}}=8^{\frac{3-4}{6}}=8^{\frac{-1}{6}}$
$(\mathrm{iii}) \frac{5^{\frac{6}{7}}}{5^{\frac{2}{3}}}=5^{\frac{6}{7}-\frac{2}{3}}=5^{\frac{18-14}{21}}=5^{\frac{4}{21}}$
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