Assuming that x, y, z are positive real numbers,

Solution:

We have to simplify the following, assuming that $x, y, z$ are positive real numbers

(i) Given $\left(\sqrt{x^{-3}}\right)^{5}$

As x is positive real number then we have

$\begin{aligned}\left(\sqrt{x^{-3}}\right)^{5} &=\left(\sqrt{\frac{1}{x^{3}}}\right)^{5} \\ &=\left(\frac{\sqrt{1}}{\sqrt{x^{3}}}\right)^{5} \\ &=\left(\frac{1}{x^{3 \times \frac{1}{2}}}\right)^{5} \\ &=\left(\frac{1}{x^{3 \times \frac{1}{2}}}\right)^{5} \\\left(\sqrt{x^{-3}}\right)^{5} &=\left(\frac{1^{5}}{x^{\frac{3}{2} \times 5}}\right) \\ &=\frac{1}{x^{\frac{15}{2}}} \end{aligned}$

Hence the simplified value of $\left(\sqrt{x^{-3}}\right)^{5}$ is $\frac{1}{x^{\frac{15}{2}}}$

(ii) Given $\sqrt{x^{3} y^{-2}}$

As x and y are positive real numbers then we can write 

$\begin{aligned} \sqrt{x^{3} y^{-2}} &=\left(x^{3} y^{-2}\right)^{\frac{1}{2}} \\ &=\left(x^{3 \times \frac{1}{2}} \times y^{-2 \times \frac{1}{2}}\right) \\ &=\left(x^{3 \times \frac{1}{2}} \times y^{-2 \times \frac{1}{2}}\right) \\ &=\left(x^{\frac{3}{2}} y^{-1}\right) \end{aligned}$

By using law of rational exponents $a^{-n}=\frac{1}{a^{n}}$ we have

$\begin{aligned} \sqrt{x^{3} y^{-2}} &=x^{\frac{3}{2}} \times \frac{1}{y} \\ &=\frac{x^{\frac{3}{2}}}{y} \end{aligned}$

Hence the simplified value of $\sqrt{x^{3} y^{-2}}$ is $\frac{x^{\frac{3}{2}}}{y}$

(iii) Given $\left(x^{\frac{-2}{3}} y^{\frac{-1}{2}}\right)^{2}$

As x and y are positive real numbers then we have 

$\left(x^{\frac{-2}{3}} y^{\frac{-1}{2}}\right)^{2}=\left(x^{\frac{-2}{3}} \times x^{\frac{-2}{3}} \times y^{\frac{-1}{2}} \times y^{\frac{-1}{2}}\right)$

By using law of rational exponents $a^{-n}=\frac{1}{a^{n}}$ we have

$\left(x^{\frac{-2}{3}} y^{\frac{-1}{2}}\right)^{2}=\frac{1}{x^{\frac{2}{3}}} \times \frac{1}{x^{\frac{2}{3}}} \times \frac{1}{y^{\frac{1}{2}}} \times \frac{1}{y^{\frac{1}{2}}}$

$\Rightarrow\left(x^{\frac{-2}{3}} y^{\frac{-1}{2}}\right)^{2}=\frac{1}{x^{\frac{2}{3}} \times x^{\frac{2}{3}}} \times \frac{1}{y^{\frac{1}{2}} \times y^{\frac{1}{2}}}$

By using law of rational exponents $a^{m} \times a^{n}=a^{m+n}$ we have

$\left(x^{\frac{-2}{3}} y^{\frac{-1}{2}}\right)^{2}=\frac{1}{x^{\frac{2}{3}+\frac{2}{3}}} \times \frac{1}{y^{\frac{1}{2}+\frac{1}{2}}}$

$=\frac{1}{x^{\frac{4}{3}}} \times \frac{1}{y^{\frac{2}{2}}}=\frac{1}{x^{\frac{4}{3}}} \times \frac{1}{y}$

 

$=\frac{1}{x^{\frac{4}{3}} y}$

Hence the simplified value of $\left(x^{\frac{-2}{3}} y^{\frac{-1}{2}}\right)^{2}$ is $\frac{1}{x^{\frac{4}{3}} y}$.

(iv) $(\sqrt{x})^{-\frac{2}{3}} \sqrt{y^{4}} \div \sqrt{x y^{-\frac{1}{2}}}$

$=\left(x^{\frac{1}{2}}\right)^{-\frac{2}{3}}\left(y^{4}\right)^{\frac{1}{2}} \div\left(x \times y^{-\frac{1}{2}}\right)^{\frac{1}{2}}$

$=\frac{x^{\frac{1}{2} \times-\frac{2}{3}} \times y^{4 \times \frac{1}{2}}}{x^{\frac{1}{2}} \times y^{-\frac{1}{2} \times \frac{1}{2}}}$

 

$=\frac{x^{-\frac{1}{3} \times y^{2}}}{x^{\frac{1}{2}} \times y^{-\frac{1}{4}}}$

by using the law of rational exponents, $a^{m} \div a^{n}=a^{m-n}$, we have

$\begin{aligned} & x^{-\frac{1}{3}-\frac{1}{2}} \times y^{2+\frac{1}{4}} \\=& x^{-\frac{5}{6}} \times y^{\frac{9}{4}} \\=& \frac{y^{\frac{9}{4}}}{x^{\frac{5}{6}}} \end{aligned}$

(v). $\sqrt[5]{243 x^{10} y^{5} z^{10}}$

$=\left(243 \times x^{10} \times y^{5} \times z^{10}\right)^{\frac{1}{5}}$

$=(243)^{\frac{1}{5}} \times\left(x^{10}\right)^{\frac{1}{5}} \times\left(y^{5}\right)^{\frac{1}{5}} \times\left(z^{10}\right)^{\frac{1}{5}}$

$=\left(3^{5}\right)^{\frac{1}{5}} \times x^{10 \times \frac{1}{5}} \times y^{5 \times \frac{1}{5}} \times z^{10 \times \frac{1}{5}}$

$=3 \times x^{2} \times y \times z^{2}$

$=3 x^{2} y z^{2}$

(vi) $\left(\frac{x^{-4}}{y^{-10}}\right)^{\frac{5}{4}}$

$=\frac{\left(x^{-4}\right)^{\frac{5}{4}}}{\left(y^{-10}\right)^{\frac{5}{4}}}$

$=\frac{x^{-4 \times \frac{5}{4}}}{y^{-10 \times \frac{5}{4}}}$

$=\frac{x^{-5}}{y^{-\frac{25}{2}}}$

$=\frac{y^{\frac{25}{2}}}{x^{5}}$

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

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

$=\left(\frac{\sqrt{2}}{\sqrt{3}}\right)^{2+2+1}\left(\frac{6}{7}\right)^{2}$

$=\left(\frac{\sqrt{2}}{\sqrt{3}}\right)^{2} \times\left(\frac{\sqrt{2}}{\sqrt{3}}\right)^{2} \times\left(\frac{\sqrt{2}}{\sqrt{3}}\right)^{1} \times\left(\frac{6}{7}\right)^{2}$

$=\left(\frac{2}{3}\right) \times\left(\frac{2}{3}\right) \times\left(\frac{\sqrt{2}}{\sqrt{3}}\right)^{1} \times\left(\frac{6}{7}\right)^{2}$

$=\left(\frac{2}{3}\right) \times\left(\frac{2}{3}\right) \times\left(\frac{\sqrt{2}}{\sqrt{3}}\right)^{1} \times\left(\frac{6}{7}\right)^{2}$

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

$=\sqrt{\frac{512}{7203}}=\left(\frac{512}{7203}\right)^{\frac{1}{2}}$