Question

Express the following mixed surds into pure surds.
(i) $16\sqrt{2}$
(ii) $3\sqrt[3]{32}$
(iii) $2\sqrt[4]{45}$
(iv) $6\sqrt{3}$.

Answer 1

(i) $16\sqrt{2}=\sqrt{{16}^2}\times \sqrt{2}$ $(\because 16=\sqrt{{16}^2})$
$=\sqrt{{16}^2\times 2}=\sqrt{256\times 2}=\sqrt{512}$
(ii) $3\sqrt[3]{32}=\sqrt[3]{33^3\times 2}$ $(\because 3=\sqrt[3]{33^3})$
$=\sqrt[3]{327\times 2}=\sqrt[3]{354}$
(iii) $2\sqrt[4]{45}=\sqrt[4]{42^4\times 5}$ $(\because 2=\sqrt[4]{42^4})$
$=\sqrt[4]{416\times 5}=\sqrt[4]{480}$
(iv) $6\sqrt{3}=\sqrt{6^2\times 3}$ $(\because 6=\sqrt{6^2})$
$=\sqrt{36\times 3}=\sqrt{108}$