Assuming that $x, y, z$ are positive real numbers, simplify each of the following:$ \left(\frac{\sqrt{2}}{\sqrt{3}}\right)^{5}\left(\frac{6}{7}\right)^{2} $

Given:

\( \left(\frac{\sqrt{2}}{\sqrt{3}}\right)^{5}\left(\frac{6}{7}\right)^{2} \)

To do:

We have to simplify the given expression.

Solution:

We know that,

$(a^{m})^{n}=a^{m n}$

$a^{m} \times a^{n}=a^{m+n}$

$a^{m} \div a^{n}=a^{m-n}$

$a^{0}=1$

Therefore,

$(\frac{\sqrt{2}}{\sqrt{3}})^{5}(\frac{6}{7})^{2}=(\sqrt{\frac{2}{3}})^5(\frac{6^2}{7^2}$

$=(\frac{2}{3})^{\frac{5}{2}}(\frac{36}{49}$

$=(\frac{2}{3})^{2} \times \frac{\sqrt{2}}{\sqrt{3}} \times \frac{36}{49}$

$=\frac{4}{9} \times \frac{\sqrt{2}}{\sqrt{3}} \times \times \frac{36}{49}$

$=\frac{16 \sqrt{2}}{49 \sqrt{3}}$

$=\frac{\sqrt{2 \times 16 \times 16}}{\sqrt{3 \times 49 \times 49}}$

$=\frac{\sqrt{512}}{\sqrt{7203}}$

$=(\frac{512}{7203})^{\frac{1}{3}}$

Hence, $(\frac{\sqrt{2}}{\sqrt{3}})^{5}(\frac{6}{7})^{2}=(\frac{512}{7203})^{\frac{1}{3}}$.

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Updated on: 10-Oct-2022

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