Question

Prove that:

(i) $\sqrt{3\times 5^{-3}}÷\sqrt[3]{3^{-1}}\sqrt{5}\times \sqrt[6]{3\times 5^6}=\frac{3}{5}$

(ii) $9^{3/2}-3\times 5^0-{{\left(\frac{1}{81}\right)}^{-}}^{1/2}=15$

(iii) ${\left(\frac{1}{4}\right)}^{-2}-3\times 8^{2/3}\times 4^0+{\left(\frac{9}{16}\right)}^{-1/2}=\frac{16}{3}$

(iv) $\frac{2^{1/2}\times 3^{1/3}\times 4^{1/4}}{{10}^{-1/5}\times 5^{3/5}}÷\frac{3^{4/3}\times 5^{-7/5}}{4^{-3/5}\times 6}=10$

(v) $\sqrt{\frac{1}{4}+(0.01{)}^{-1/2}}-(27{)}^{2/3}=\frac{3}{2}$

(vi) $\frac{2^n+2^{n-1}}{2^{n+1}-2^n}=\frac{3}{2}$

(vii) ${\left(\frac{64}{125}\right)}^{-2/3}+\frac{1}{{\left({\displaystyle \frac{256}{625}}\right)}^{1/4}}+\left(\frac{\sqrt{25}}{\sqrt[3]{64}}\right)=\frac{65}{16}$

(viii) $\frac{3^{-3}\times 6^2\times \sqrt{98}}{5^2\times \sqrt[3]{1/25}\times \left(15\right){-}^{4/3}\times 3^{1/3}}=28\sqrt{2}$

(ix) $\frac{(0.6{)}^0-(0.1)-1}{{\left({\displaystyle \frac{3}{8}}\right)}^{-1}{\left({\displaystyle \frac{3}{2}}\right)}^3+{\left(-{\displaystyle \frac{1}{3}}\right)}^{-1}}=-\frac{3}{2}$

Answer 1

(i) We have to prove that

By using rational exponent we get,

Hence,

(ii) We have to prove that. So,

Hence,

(iii) We have to prove that

Now,

Hence,

(iv) We have to prove that. So,

Let

Hence,

(v) We have to prove that

Let

Hence,

(vi) We have to prove that . So,

Let

Hence,

(vii) We have to prove that. So let

By taking least common factor we get

Hence,

(viii) We have to prove that. So,

Let

Hence,

(ix) We have to prove that. So,

Let

Hence,