If $a=x{y}^{p-1},b=x{y}^{q-1}$ and $c=x{y}^{r-1}$ prove that $a^{q-r}{b}^{r-p}{c}^{p-q}=1$

Assuming that $x$ is a positive real numbers, simplify the following:

$(\sqrt{x^{-3}}{)}^5$

Assuming that $x, y$ are positive real numbers, simplify the following:

$\sqrt{x^3{y}^{-2}}$

Assuming that $x, y$ are positive real numbers, simplify the following:

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

Assuming that $x, y$ are positive real numbers, simplify the following:

${\left(\sqrt{x}\right)}^{-\frac{2}{3}}\sqrt{y^4}÷\sqrt{x{y}^{-\frac{1}{2}}}$

Assuming that $x, y, z$ are positive real numbers, simplify the following:

$\sqrt[5]{243x^{10}{y}^5{z}^{10}}$

Assuming that $x, y, z$ are positive real numbers, simplify the following:

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

Simplify the following:

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

If we simplify the following, we get $\frac{a}{b}$, find $a+b$

${\left({16}^{-\frac{1}{5}}\right)}^{\frac{5}{2}}$