Question

$\sqrt[3]{2}\times \sqrt[4]{2}\times \sqrt[12]{32}=?$

(a) 2

(b) $\sqrt{2}$

(c) $2\sqrt{2}$

(d) $4\sqrt{2}$

Answer 1

$$\begin{gathered} \sqrt[3]{2}\times \sqrt[4]{2}\times \sqrt[12]{32}={\left(2\right)}^{\frac{1}{3}}\times {\left(2\right)}^{\frac{1}{4}}\times {\left(32\right)}^{\frac{1}{12}} \\ ={\left(2\right)}^{\frac{1}{3}}\times {\left(2\right)}^{\frac{1}{4}}\times {\left(2^5\right)}^{\frac{1}{12}} \\ ={\left(2\right)}^{\frac{1}{3}}\times {\left(2\right)}^{\frac{1}{4}}\times {\left(2\right)}^{\frac{5}{12}} \\ ={\left(2\right)}^{\frac{1}{3}+\frac{1}{4}+\frac{5}{12}} \\ ={\left(2\right)}^{\frac{4+3+5}{12}} \\ ={\left(2\right)}^{\frac{12}{12}} \\ =2 \\ \therefore \sqrt[3]{2}\times \sqrt[4]{2}\times \sqrt[12]{32}=2. \end{gathered}$$

Hence, the correct option is (a).