Question
Simplify the following:$\sqrt[3]{x^4 y^2} \div \sqrt[6]{x^5 y^{-5}}$
✓
Answer
$ \sqrt[3]{x^4 y^2} \div \sqrt[6]{x^5 y^{-5}}$
$=\left(x^4 y^2\right)^{\frac{1}{3}} \div\left(x^5 y^{-5}\right)^{\frac{1}{6}}$
$=\left(x^{4 \times \frac{1}{3}} y^{2 \times \frac{1}{3}}\right) \div\left(x^{5 \times \frac{1}{6}} y^{-5 \times \frac{1}{6}}\right) \quad \ldots($Using $\left( a ^{ m }\right)^n= a ^{ mn })$
$=\left(x^{\frac{4}{3}} y^{\frac{2}{3}}\right) \div\left(x^{\frac{5}{6}} y^{-\frac{5}{6}}\right)$
$=\frac{x^{\frac{4}{3}} y^{\frac{2}{3}}}{x^{\frac{5}{6}} y^{-\frac{5}{6}}}$
$=x^{\frac{4}{3}-\frac{5}{6}} y^{\frac{2}{3}-\left(-\frac{5}{6}\right)} \ldots($Using $\left( a ^{ m }\right)^{ n }= a ^ {mn})$
$=x^{\frac{1}{2}} y^{\frac{3}{2}}$
$=x^{\frac{1}{2}}\left(y^3\right)^{\frac{1}{2}}$
$=\sqrt{x} \sqrt{y^3}$
$=\sqrt{x y^3} . \ldots($Using $\left.\left(a^m\right)^n=a^{m n}\right)$
$=\left(x^4 y^2\right)^{\frac{1}{3}} \div\left(x^5 y^{-5}\right)^{\frac{1}{6}}$
$=\left(x^{4 \times \frac{1}{3}} y^{2 \times \frac{1}{3}}\right) \div\left(x^{5 \times \frac{1}{6}} y^{-5 \times \frac{1}{6}}\right) \quad \ldots($Using $\left( a ^{ m }\right)^n= a ^{ mn })$
$=\left(x^{\frac{4}{3}} y^{\frac{2}{3}}\right) \div\left(x^{\frac{5}{6}} y^{-\frac{5}{6}}\right)$
$=\frac{x^{\frac{4}{3}} y^{\frac{2}{3}}}{x^{\frac{5}{6}} y^{-\frac{5}{6}}}$
$=x^{\frac{4}{3}-\frac{5}{6}} y^{\frac{2}{3}-\left(-\frac{5}{6}\right)} \ldots($Using $\left( a ^{ m }\right)^{ n }= a ^ {mn})$
$=x^{\frac{1}{2}} y^{\frac{3}{2}}$
$=x^{\frac{1}{2}}\left(y^3\right)^{\frac{1}{2}}$
$=\sqrt{x} \sqrt{y^3}$
$=\sqrt{x y^3} . \ldots($Using $\left.\left(a^m\right)^n=a^{m n}\right)$
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