Evaluate: $\frac{6^{\frac{2}{3}} \times \sqrt[3]{6^{7}}}{\sqrt[3]{6^{6}}}$

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Solution

We have
$\frac{6^{\frac{2}{3}} \times \sqrt[3]{6^{7}}}{\sqrt[3]{6^{6}}}$
$= \frac{6^{\frac{2}{3}} \times 6^{\frac{7}{3}}}{6^{\frac{6}{3}}}$ [ $∵ a^{m} \times a^{n} = a^{(m + n)}$ , $\sqrt{a} = a^{\frac{1}{a}}, a^{m} \div b^{n} = a^{m - n}$ ]
$= 6^{(\frac{2}{3} + \frac{7}{3} - \frac{6}{3})}$
$= 6^{(\frac{3}{3})}$
$= 6^{1}$
$= 6$

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