Question
Simplify the following:$\left(\frac{64 a^{12}}{27 b^6}\right)^{-\frac{2}{3}}$
✓
Answer
$\left(\frac{64 a^{12}}{27 b^6}\right)^{-\frac{2}{3}}$
$=\left(\frac{2^6 a^{12}}{3^3 b^6}\right)^{-\frac{2}{3}}$
$=\left(\frac{2^{6 \times\left(-\frac{2}{3}\right)} a^{12 \times\left(-\frac{2}{3}\right)}}{3^{3 \times\left(-\frac{2}{3}\right)} b^{6 \times\left(-\frac{2}{3}\right)}}\right) \ldots \ldots($Using $\left(a \times b^n\right)=a^n \times b^n$ and $\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n})$
$=\frac{2^{-4} a^{-8}}{3^{-2} b^{-4}}$
$=\frac{3^2 b^4}{2^4 a^8} \cdots \cdot($Using $a^{-n}=\frac{1}{a^n})$
$=\frac{9 b^4}{16 a^8} .$
$=\left(\frac{2^6 a^{12}}{3^3 b^6}\right)^{-\frac{2}{3}}$
$=\left(\frac{2^{6 \times\left(-\frac{2}{3}\right)} a^{12 \times\left(-\frac{2}{3}\right)}}{3^{3 \times\left(-\frac{2}{3}\right)} b^{6 \times\left(-\frac{2}{3}\right)}}\right) \ldots \ldots($Using $\left(a \times b^n\right)=a^n \times b^n$ and $\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n})$
$=\frac{2^{-4} a^{-8}}{3^{-2} b^{-4}}$
$=\frac{3^2 b^4}{2^4 a^8} \cdots \cdot($Using $a^{-n}=\frac{1}{a^n})$
$=\frac{9 b^4}{16 a^8} .$
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