Question

Prove that :

$\left(i\right)(\sqrt{3\times 5^{-3}}÷\sqrt[3]{33^{-1}}\sqrt{5})\times \sqrt[6]{63\times 5^6}=\frac{3}{5}\left(ii\right)9^{\frac{3}{2}}-3\times 5^0-{\left(\frac{1}{81}\right)}^{\frac{-1}{2}}=15\left(iii\right){\left(\frac{1}{4}\right)}^{-2}-3\times 8^{\frac{2}{3}}\times 4^0+{\left(\frac{9}{16}\right)}^{\frac{-1}{2}}=\frac{16}{3}\left(iv\right)\frac{2^{\frac{1}{2}}\times 3^{\frac{1}{3}}\times 4^{\frac{1}{4}}}{{10}^{\frac{-1}{5}}\times 5^{\frac{3}{5}}}÷\frac{3^{\frac{4}{3}}\times 5^{\frac{-7}{5}}}{4^{\frac{-3}{5}}\times 6}=10\left(v\right)\sqrt{\frac{1}{4}}+(0.01{)}^{\frac{-1}{2}}-(27{)}^{\frac{2}{3}}=\frac{3}{2}\left(vi\right)\frac{2^n+2^{n-1}}{2^{n+1}-2^n}=\frac{3}{2}\left(vii\right){\left(\frac{64}{125}\right)}^{\frac{-2}{3}}+\frac{1}{{\left(\frac{256}{625}\right)}^{\frac{1}{4}}}+{\left(\frac{\sqrt{25}}{\sqrt[3]{364}}\right)}^0=\frac{61}{16}\left(viii\right)\frac{3^{-3}\times 6^2\times \sqrt{98}}{5^2\times \sqrt[3]{3\frac{1}{25}}\times (15{)}^{\frac{-4}{3}}\times 3^{\frac{1}{3}}}=28\sqrt{2}\left(ix\right)\frac{(0.6{)}^0-(0.1{)}^{-1}}{{\left(\frac{3}{8}\right)}^{-1}{\left(\frac{3}{2}\right)}^3+{\left(\frac{1}{3}\right)}^{-1}}=-\frac{3}{2}$

Answer 1

$\left(i\right)\sqrt{3\times 5^{-3}}÷\sqrt[3]{33^{-1}}\sqrt{5}\times \sqrt[6]{63\times 5^6}=\frac{3}{5}L.H.S=\sqrt{3\times 5^{-3}}÷\sqrt[3]{33^{-1}}\sqrt{5}\times \sqrt[6]{63\times 5^6}=(3\times 5^{-3}{)}^{\frac{1}{2}}÷(3^{-1}{)}^{\frac{1}{3}}(5{)}^{\frac{1}{2}}\times (3\times 5^6{)}^{\frac{1}{6}}=(3^{\frac{1}{2}}\times 5^{\frac{-3}{2}})÷(3^{\frac{-1}{3}}\times 5^{\frac{1}{2}})\times (3^{\frac{1}{6}}\times 5^{6\times \frac{1}{6}})=(3^{\frac{1}{2}}\times 5^{\frac{-3}{2}})÷(3^{\frac{-1}{3}}\times 5^{\frac{1}{2}})\times 3^{\frac{1}{6}}\times 5^1=3^{\frac{1}{2}-\left(\frac{-1}{3}\right)+\frac{1}{6}{.5}^{\frac{-3}{2}-\frac{1}{2}+1}}=3^{\frac{1}{2}+\frac{1}{3}+\frac{1}{6}{.5}^{\frac{-3-1+2}{2}}}=3^{\frac{3+2+1}{6}{.5}^{\frac{-4+2}{2}}}=3^{\frac{6}{6}\times 5^{\frac{-2}{2}}}=3^1{.5}^{-1}=\frac{3}{5}=R.H.S$

$\left(ii\right)9^{\frac{3}{2}}-3\times 5^0-{\left(\frac{1}{81}\right)}^{\frac{-1}{2}}=15L.H.S=9^{\frac{3}{2}}-3\times 5^0-{\left(\frac{1}{81}\right)}^{\frac{-1}{2}}=(3^2{)}^{\frac{3}{2}}-3\times 5^0-{\left(\frac{1}{9^2}\right)}^{\frac{-1}{2}}\{\because a^0=1\}=3^{2\times \frac{3}{2}}-3-{\left(\frac{1}{9}\right)}^{2\times \left(\frac{-1}{2}\right)}=3^3-3-{\left(\frac{1}{9}\right)}^{-1}27-3-{\left(\frac{1}{9}\right)}^{-1}27-3-9=15=R.H.S.$

$\left(iii\right){\left(\frac{1}{4}\right)}^{-2}-3\times 8^{\frac{2}{3}}\times 4^0+{\left(\frac{9}{16}\right)}^{\frac{-1}{2}}=\frac{16}{3}L.H.S={\left(\frac{1}{4}\right)}^{-2}-3\times 8^{\frac{2}{3}}\times 4^0+{\left(\frac{9}{16}\right)}^{\frac{-1}{2}}={\left[{\left(\frac{1}{2}\right)}^2\right]}^{-2}-3\times (2^3{)}^{\frac{2}{3}}\times 1+{\left(\frac{3}{4}\right)}^{2\times \frac{-1}{2}}={\frac{1}{2}}^{2\times \frac{-1}{2}}-3\times 2^{3\times \frac{2}{3}}\times 1+{\left(\frac{3}{4}\right)}^{2\times \left(\frac{-1}{2}\right)}={\frac{1}{2}}^{-4}-3\times 2^2\times 1+{\left(\frac{3}{4}\right)}^{-1}={\left(\frac{2}{1}\right)}^4-3\times 4+\frac{4}{2}\{\because a^{-m}=\frac{1}{a^m}\}={\left(\frac{2}{1}\right)}^4-12+\frac{4}{3}=16-12+\frac{4}{3}=4+\frac{4}{3}=\frac{16}{3}=R.H.S$

(iv)$\frac{2^{\frac{1}{2}}\times 3^{\frac{1}{3}}\times 4^{\frac{1}{4}}}{{10}^{\frac{-1}{5}}\times 5^{\frac{3}{5}}}÷\frac{3^{\frac{4}{3}}\times 5^{\frac{-7}{5}}}{4^{\frac{-3}{5}}\times 6}=10L.H.S.=\frac{2^{\frac{1}{2}}\times 3^{\frac{1}{3}}\times 4^{\frac{1}{4}}}{{10}^{\frac{-1}{5}}\times 5^{\frac{3}{5}}}÷\frac{3^{\frac{4}{3}}\times 5^{\frac{-7}{5}}}{4^{\frac{-3}{5}}\times 6}=\frac{2^{\frac{1}{2}}\times 3^{\frac{1}{3}}\times (2^2{)}^{\frac{1}{4}}}{(2\times 5{)}^{\frac{-1}{5}}\times 5^{\frac{3}{5}}}÷\frac{3^{\frac{4}{3}}\times 5^{\frac{-7}{5}}}{(2^2{)}^{\frac{-3}{5}}\times 2^1\times 3^1}=\frac{2^{\frac{1}{2}}\times 3^{\frac{1}{3}}\times 2^{2\times \frac{1}{4}}}{2^{\frac{-1}{5}\times }{5}^{\frac{-1}{5}\times }{5}^{\frac{3}{5}}}÷\frac{3^{\frac{4}{3}}\times 5^{\frac{-7}{5}}}{2^{\frac{-6}{5}}\times 2^1\times 3^1}=\frac{2^{\frac{1}{2}}\times 3^{\frac{1}{3}}\times 2^{2\times \frac{1}{4}}}{2^{\frac{-1}{5}\times }{5}^{\frac{-1}{5}\times }{5}^{\frac{3}{5}}}\times \frac{2^{\frac{-6}{5}}\times 2^1\times 3^1}{3^{\frac{4}{3}}\times 5^{\frac{-7}{5}}}=2^{\frac{1}{2}+\frac{1}{2}-\frac{6}{5}+1+\frac{1}{5}}{.3}^{\frac{1}{3}+1-\frac{4}{3}}{.5}^{\frac{1}{5}-\frac{3}{5}+\frac{7}{5}}==2^{\frac{5+5-12+10+2}{10}}{.3}^{\frac{1+3-4}{3}}{.5}^{\frac{1-3+7}{5}}=2^{\frac{22-12}{10}}{.3}^{4-4}{.5}^{\frac{8-3}{5}}=2^{\frac{10}{10}}{.3}^0{.5}^{\frac{5}{5}}=2^1{.3}^0{.5}^1=2\times 1\times 5=10=R.H.S.$

$\left(v\right)\sqrt{\frac{1}{4}}+(0.01{)}^{\frac{-1}{2}}-(27{)}^{\frac{2}{3}}=\frac{3}{2}L.H.S.=\sqrt{\frac{1}{4}}+(0.01{)}^{\frac{-1}{2}}-(27{)}^{\frac{2}{3}}={\left(\frac{1}{2^2}\right)}^{\frac{1}{2}}+(0.1{)}^{2\times \left(\frac{-1}{2}\right)}-(3^3{)}^{\frac{2}{3}}=\frac{1}{(2^2{)}^{\frac{1}{2}}}+(0.1{)}^{2\times \frac{-1}{2}}-3^{3\times \frac{2}{3}}=\frac{1}{2^1}+(0.1{)}^{-1}-3^2=\frac{1}{2}+{\left(\frac{10}{1}\right)}^1-3^2=\frac{1}{2}+10-9=1\frac{1}{2}=\frac{3}{2}=R.H.S.$

$\left(vi\right)\frac{2^n+2^{n-1}}{2^{n-1}-2^n}=\frac{3}{2}L.H.S=\frac{2^n+2^{n-1}}{2^{n-1}-2^n}=\frac{2^n+2^n\times 2^{-1}}{2^n{.2}^1-2^n}=\frac{2^n(1+2^{-1})}{2^n(2^1-1)}=\frac{1+\frac{1}{2}}{2-1}=\frac{\frac{3}{2}}{1}=\frac{3}{2}=R.H.S.$

$\left(vii\right){\left(\frac{64}{125}\right)}^{\frac{-2}{3}}+\frac{1}{{\left(\frac{256}{625}\right)}^{\frac{1}{4}}}+{\left(\frac{{\sqrt{25}}^0}{\sqrt[3]{364}}\right)}^0=\frac{61}{16}L.H.S.={\left(\frac{64}{125}\right)}^{\frac{-2}{3}}+\frac{1}{{\left(\frac{256}{625}\right)}^{\frac{1}{4}}}+\left(\frac{\sqrt{25}}{\sqrt[3]{364}}\right)={\left(\frac{4^3}{5^3}\right)}^{\frac{-2}{3}}+\frac{1}{{\left(\frac{4^4}{5^4}\right)}^{\frac{1}{4}}}+\left(\frac{\sqrt{5^2}}{\sqrt[3]{34^3}}\right)=\frac{4^{3\times \left(\frac{-2}{3}\right)}}{5^3\times \frac{-2}{3}}+\frac{1}{\frac{4^{4\times \frac{1}{4}}}{5^{4\times \frac{1}{4}}}}+\frac{(5^2{)}^{\frac{1}{2}}}{(4^3{)}^{\frac{1}{3}}}=\frac{5^2}{4^2}+\frac{5}{4}+\frac{5}{4}=\frac{25}{16}+\frac{10}{4}=\frac{25}{16}+\frac{40}{16}=\frac{65}{16}=R.H.S.$

$\left(viii\right)\frac{3^{-3}\times 6^2\times \sqrt{98}}{5^2\times \sqrt[3]{3\frac{1}{25}}\times (15{)}^{\frac{-4}{3}}\times 3^{\frac{1}{3}}}=28\sqrt{2}L.H.S.=\frac{3^{-3}\times 6^2\times \sqrt{98}}{5^2\times \sqrt[3]{3\frac{1}{25}}\times (15{)}^{\frac{-4}{3}}\times 3^{\frac{1}{3}}}=\frac{3^{-3}\times (2^2\times 3^2)\times (2\times 49{)}^{\frac{1}{2}}}{5^2\times (25{)}^{\frac{-1}{3}}\times 3^{\frac{-4}{3}}\times 5^{\frac{-4}{3}}\times 3^{\frac{1}{3}}}=\frac{3^{-3}\times 2^2\times 3^2\times 2^{\frac{1}{2}}\times (7^2{)}^{\frac{1}{2}}}{5^2\times (5^2{)}^{\frac{-1}{3}}\times 3^{\frac{-4}{3}}\times 5^{\frac{-4}{3}}\times 3^{\frac{1}{3}}}=2^2{.2}^{\frac{1}{2}}{.3}^{-3+2+\frac{4}{3}-\frac{1}{3}}{.5}^{\frac{2}{3}-2+\frac{4}{3}}{.7}^1=4\sqrt{2}\times 3^0{.5}^0\times 7^1=4\sqrt{2}\times 1\times 1\times 7=28\sqrt{2}=R.H.S.$

$\left(ix\right)\frac{(0.6{)}^0-(0.1{)}^{-1}}{{\left(\frac{3}{8}\right)}^{-1}{\left(\frac{3}{2}\right)}^3+{(-\frac{1}{3})}^{-1}}=-\frac{3}{2}L.H.S=\frac{(0.6{)}^0-(0.1{)}^{-1}}{{\left(\frac{3}{8}\right)}^{-1}{\left(\frac{3}{2}\right)}^3+{(-\frac{1}{3})}^{-1}}=\frac{1-{10}^1}{\frac{3}{8}\times \frac{27}{8}+\left(\frac{-3}{1}\right)}=\frac{-9}{9-\frac{3}{1}}=\frac{-9}{6}=\frac{-3}{2}=RHS$