Question

Simplify:

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

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

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

(iv) $(0.001{)}^{\frac{1}{3}}$

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

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

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

Answer 1

(i) $({16}^{\frac{-1}{5}}{)}^{\frac{5}{2}}={\left[{\left(2^4\right)}^{\frac{-1}{5}}\right]}^{\frac{5}{2}}$ =$2^{4\times \frac{-1}{5}\times \frac{5}{2}}\left\{\begin{array}{cc}\because (a^m{)}^n=a^{mn}& \\ a^{-m}=\frac{1}{a^m}& \end{array}\right\}=2^{-2}=\frac{1}{2^2}=\frac{1}{2\times 2}=\frac{1}{4}$

(ii) $\sqrt[5]{5(32{)}^{-3}}=\sqrt[5]{5\frac{1}{{32}^3}}=\frac{1}{{32}^{\frac{3}{5}}}=\frac{1}{(2^5{)}^{\frac{3}{5}}}=\frac{1}{2^{5\times \frac{3}{5}}}=\frac{1}{2^3}=\frac{1}{2\times 2\times 2}=\frac{1}{8}$

(iii) $\sqrt[3]{3(343{)}^{-2}}=\sqrt[3]{3\frac{1}{{343}^2}}=\frac{1}{{343}^{\frac{2}{3}}}=\frac{1}{(7^3{)}^{\frac{2}{3}}}=\frac{1}{7^{3\times \frac{2}{3}}}=\frac{1}{7^2}=\frac{1}{7\times 7}=\frac{1}{49}$

(iv) $(0.001{)}^{\frac{1}{3}}={\left\{\right(0.1{)}^3\}}^{\frac{1}{3}}=(0.1{)}^{3\times \frac{1}{3}}=(0.1{)}^1=0.1$

(v) $\frac{(25{)}^{\frac{3}{2}}\times (243{)}^{\frac{3}{5}}}{(16{)}^{\frac{5}{4}}\times (8{)}^{\frac{4}{3}}}=\frac{(5^2{)}^{\frac{3}{2}}\times (3^5{)}^{\frac{3}{5}}}{(2^4{)}^{\frac{5}{4}}\times (2^3{)}^{\frac{4}{3}}}=\frac{5^{2\times \frac{3}{2}\times }{3}^{5\times \frac{3}{5}}}{2^{4\times \frac{5}{4}}\times 2^{3\times \frac{4}{3}}}=\frac{5^3\times 3^3}{2^5\times 2^4}=\frac{5\times 5\times 5\times 3\times 3\times 3}{2\times 2\times 2\times 2\times 2\times 2\times 2\times 2\times 2}=\frac{3375}{512}$

(vi) ${\left(\frac{\sqrt{2}}{5}\right)}^8÷{\left(\frac{\sqrt{2}}{5}\right)}^{13}={\left(\frac{\sqrt{2}}{5}\right)}^{8-13}={\left(\frac{\sqrt{2}}{5}\right)}^{-5}=\frac{(\sqrt{2}{)}^{-5}}{5^{-5}}=\frac{5^5}{(\sqrt{2}{)}^5}=\frac{5^5}{2^{\frac{5}{2}}}=\frac{5\times 5\times 5\times 5\times 5}{\sqrt{2\times 2\times 2\times 2\times 2}}=\frac{3125}{2\times 2\times \sqrt{2}}=\frac{3125}{4\sqrt{2}}$

(vii) ${\left(\frac{5^{-1}\times 7^2}{5^2\times 7^{-4}}\right)}^{\frac{7}{2}}\times {\left(\frac{5^{-2}\times 7^3}{5^3\times 7^{-5}}\right)}^{\frac{-5}{2}}=\frac{5^{-1\times \frac{7}{2}}\times 7^{2\times \frac{7}{2}}}{5^{2\times \frac{7}{2}}\times 7^{-4\times \frac{7}{2}}}\times \frac{5^{-2\times \left(\frac{-5}{2}\right)}\times 7^{3\times \left(\frac{-5}{2}\right)}}{5^{3\times \left(\frac{-5}{2}\right)}\times 7^{-5\times \left(\frac{-5}{2}\right)}}=\frac{5^{\frac{-7}{2}}\times 7^7}{5^7\times 7^{-14}}\times \frac{5^5\times 7^{\frac{-15}{2}}}{5^{\frac{-15}{2}}\times 7^{\frac{25}{2}}}=5^{\frac{-7}{2}+5-7+\frac{15}{2}}\times 7^{7-\frac{15}{2}+14-\frac{25}{2}}=5^2\times 7^1=25\times 7=175$