Question

Question no 4 1st part and question no 5??

Q4. Simplify

(i) $∛2∛32$ (ii) $∛54$ (iii) $\sqrt[3]{\sqrt[4]{2}}$

Q5. Which is greater $∛64\mathrm{or}∜128$ ?

Answer 1

Dear student
$$\begin{gathered} \sqrt[3]{2}\times \sqrt[3]{32} \\ =2^{\frac{1}{3}}\times {32}^{\frac{1}{3}}\left[\because \sqrt[\mathrm{n}]{\mathrm{a}}={\mathrm{a}}^{1/\mathrm{n}}\right] \\ =2^{\frac{1}{3}}\times {\left(2^5\right)}^{\frac{1}{3}} \\ =2^{\frac{1}{3}}\times 2^{\frac{5}{3}}\left[\because {\left({\mathrm{a}}^{\mathrm{n}}\right)}^{\mathrm{m}}={\mathrm{a}}^{\mathrm{mn}}\right] \\ =2^{\frac{1}{3}+\frac{5}{3}}\left[\because {\mathrm{a}}^{\mathrm{m}}\times {\mathrm{a}}^{\mathrm{n}}={\mathrm{a}}^{\mathrm{m}+\mathrm{n}}\right] \\ =2^{\frac{6}{3}} \\ =2^2 \\ =4 \end{gathered}$$
Regards