Assuming that x, y, z are positive real numbers, simplify each of the following
Question:

Assuming that x, y, z are positive real numbers, simplify each of the following

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

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

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

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

(v) $\sqrt[5]{243 \times 10^{5} z^{10}}$

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

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

Solution:

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

$(\sqrt{(x-3)})^{5}=\left(\sqrt{\frac{1}{x^{3}}}\right)^{5}$

$\left(\frac{1}{x \frac{3}{2}}\right)^{5}=\frac{1}{x \frac{15}{2}}$

$(\sqrt{(x-3)})^{5}=\frac{1}{x \frac{15}{2}}$

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

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

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

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

$=\frac{x \frac{3}{2}}{y}$

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

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

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

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

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

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

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

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

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

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

$=\left(x \frac{-2-3}{6}\right)\left(y^{\frac{8+1}{4}}\right)$

$=\left(x^{-\frac{5}{6}}\right)\left(y^{-\frac{9}{4}}\right)$

$=\frac{y^{\frac{9}{4}}}{x^{\frac{5}{6}}}$

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

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

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

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

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

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

$=\left(\frac{y^{\frac{25}{2}}}{x^{5}}\right)$

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

$=\left(\sqrt{\frac{2}{3}}\right)^{5}\left(\frac{6}{7}\right)^{\frac{4}{2}}$

$=\left(\frac{2}{3}\right)^{\frac{5}{2}}\left(\frac{6}{7}\right)^{\frac{4}{2}}$

$=\left(\frac{2^{5}}{3^{5}}\right)^{\frac{1}{2}}\left(\frac{6^{4}}{7^{4}}\right)^{\frac{1}{2}}$

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

$=\left(\frac{512}{7203}\right)$