Question

Find the product.
(i) $\sqrt[3]{2}\times \sqrt[3]{3}\times \sqrt[3]{4}$

(ii) $\sqrt[3]{3}\times \sqrt{3}\times \sqrt[3]{2}$

(iii) $\sqrt[3]{2}\times \sqrt[4]{2}\times \sqrt{2}$

(iv) $2\times \sqrt[3]{4}\times 3\times \sqrt[3]{16}$

(v) $3\times \sqrt[3]{32}\times 3\times \sqrt{4}$

Answer 1

$$\begin{gathered} \left(\mathrm{i}\right)\sqrt[3]{2}\times \sqrt[3]{3}\times \sqrt[3]{4} \\ =\sqrt[3]{2\times 3\times 4} \\ =\sqrt[3]{24} \\ =\sqrt[3]{8\times 3} \\ =\sqrt[3]{8}\times \sqrt[3]{3} \\ =2\sqrt[3]{3} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{ii}\right)\sqrt[3]{3}\times \sqrt[]{3}\times \sqrt[3]{2} \\ =\sqrt[3\times 2]{3^2}\times \sqrt[2\times 3]{3^3}\times \sqrt[3\times 2]{2^2} \\ =\sqrt[6]{9}\times \sqrt[6]{27}\times \sqrt[6]{4} \\ =\sqrt[6]{9\times 27\times }4 \\ =\sqrt[6]{972} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iii}\right)\sqrt[3]{2}\times \sqrt[4]{2}\times \sqrt[]{2} \\ =\sqrt[3\times 4]{2^4}\times \sqrt[4\times 3]{2^3}\times \sqrt[2\times 6]{2^6} \\ =\sqrt[12]{16}\times \sqrt[12]{8}\times \sqrt[12]{64} \\ =\sqrt[12]{16\times 8\times 64} \\ =\sqrt[12]{8192} \\ =\sqrt[12]{4096\times 2} \\ =\sqrt[12]{4096}\times \sqrt[12]{2} \\ =2\sqrt[12]{2} \end{gathered}$$

$$\begin{gathered} \left(\mathrm{iv}\right)2\times \sqrt[3]{4}\times 3\times \sqrt[3]{16} \\ =2\times 3\times \sqrt[3]{4\times 16} \\ =6\sqrt[3]{64} \\ =6\times 4 \\ =24 \end{gathered}$$

$$\begin{gathered} \left(\mathrm{v}\right)3\times \sqrt[3]{32}\times 3\times \sqrt[]{4} \\ =3\times 3\times \sqrt[3]{8}\times \sqrt[3]{4}\times 2 \\ =9\times 2\times 2\times \sqrt[3]{4} \\ =36\sqrt[3]{4} \end{gathered}$$