# Simplify

Question:

Simplify

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

(ii) $\sqrt[5]{(32)^{-3}}$

(iii) $\sqrt[3]{(343)^{-2}}$

(iv) $\frac{(25)^{\frac{3}{2}} \times(243)^{\frac{3}{5}}}{(16)^{\frac{5}{4}} \times(8)^{\frac{4}{3}}}$

(v) $\left(\frac{\sqrt{2}}{5}\right)^{8} \div\left(\frac{\sqrt{2}}{5}\right)^{13}$

(vi) $\left[\frac{5^{-1} \times 7^{2}}{5^{2} \times 7^{-4}}\right]^{\frac{7}{2}} \times\left[\frac{5^{-2} \times 7^{3}}{5^{3} \times 7^{-5}}\right]^{\frac{-5}{2}}$

Solution:

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

$=(16)^{-\frac{1}{5} \times \frac{5}{2}}$

$=(16)^{-\frac{1}{2}}$

$=\left(4^{2}\right)^{-\frac{1}{2}}$

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

$=\left(4^{-1}\right)$

$=1 / 4$

(ii) $\sqrt[5]{(32)^{-3}}$

$\sqrt[5]{(32)^{-3}}$

$=\left[\left(2^{5}\right)^{-3}\right] 1 / 5$

$=\left(2^{-15}\right)^{1 / 5}$

$=2^{-3}$

$=1 / 2^{3}=1 / 8$

(iii) $\sqrt[3]{(343)^{-2}}$

$=\left[(343)^{-2}\right]^{1 / 3}$

$=(343)^{-2 \times 1 / 3}$

$=\left(7^{3}\right)^{-2 / 3}$

$=\left(7^{-2}\right)$

$=\left(1 / 7^{2}\right)$

$=(1 / 49)$

(iv) $(0.001)^{1 / 3}$

$=(1 / 1000)^{1 / 3}$

$=\left(1 / 10^{3}\right)^{1 / 3}$

$=\left(\frac{1^{\frac{1}{3}}}{\left(10^{3}\right)^{\frac{1}{3}}}\right)$

$\frac{1}{10^{3} \times \frac{1}{3}}$

$=1 / 10=0.1$

(v) $\frac{(25)^{\frac{3}{2}} \times(243)^{\frac{3}{5}}}{(16)^{\frac{5}{4}} \times(8)^{\frac{4}{3}}}$

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

$=\frac{5^{2} \times \frac{3}{2} \times 3^{5} \times \frac{3}{5}}{2^{4} \times \frac{5}{4} \times 2^{3} \times \frac{4}{3}}$

$=\frac{5^{3} \times 3^{3}}{2^{5} \times 2^{4}}$

$=\frac{125 \times 27}{32 \times 16}$

$=\frac{3375}{512}$

(vi) $\left(\frac{\sqrt{2}}{5}\right)^{8} \div\left(\frac{\sqrt{2}}{5}\right)^{13}$

$=\frac{\left(\frac{\sqrt{2}}{5}\right)^{8}}{\left(\frac{\sqrt{2}}{5}\right)^{13}}$

$=\left(\frac{\sqrt{2}}{5}\right)^{8-13}=\left(\frac{\sqrt{2}}{5}\right)^{-5}$

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

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

$=\frac{1}{2 \frac{5}{2}} \times \frac{5^{5}}{1}$

$=\frac{5^{5}}{2 \frac{5}{2}}=\frac{3125}{4 \sqrt{2}}$

(vii) $\left[\frac{5^{-1} \times 7^{2}}{5^{2} \times 7^{-4}}\right]^{\frac{7}{2}} \times\left[\frac{5^{-2} \times 7^{3}}{5^{3} \times 7^{-5}}\right]^{\frac{-5}{2}}$

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

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

$=\frac{5^{-\frac{7}{2}} \times 7^{7}}{5^{7} \times 7^{-14}} \times \frac{5^{5} \times 7^{-\frac{15}{2}}}{5^{-\frac{15}{2}} \times 7^{-\frac{25}{2}}}$

$=\frac{7^{7}+\frac{7}{14}}{5^{7}+\frac{7}{2}} \times \frac{5^{5}+\frac{15}{2}}{7 \frac{15}{2}+\frac{25}{2}}$

$=\frac{7^{21}}{5^{\frac{21}{2}}} \times \frac{5 \frac{25}{2}}{7 \frac{40}{2}}$

$=\frac{721}{720} \times \frac{5^{\frac{25}{2}}}{5^{\frac{21}{2}}}$

$=7^{21-20} \times 5^{25 / 2-21 / 2}$

$=7^{1} \times 5^{4 / 2}$

$=7^{1} \times 5^{2}$

$=7 \times 25$

$=175$