Question

Simplify and express the result in its simplest form.
(i) $\sqrt[3]{34}\times \sqrt[3]{322}$ (ii) $\sqrt[4]{412}\times \sqrt[7]{76}$

(iii)$\sqrt[3]{332}\times \sqrt[3]{3250}$ (iv) $\sqrt[5]{532}\times 2\sqrt[3]{381}$

Answer 1

(i) $\sqrt[3]{34}\times \sqrt[3]{322}=\sqrt[3]{34\times 22}$
$=\sqrt[3]{32\times 2\times 2\times 11}=\sqrt[3]{32^3\times 11}$
$=\sqrt[3]{32^3}\sqrt[3]{311}=2\sqrt[3]{311}$
(ii) $4\sqrt{12}\times 7\sqrt{6}$
$=\sqrt{4^2}\times \sqrt{12}\times \sqrt{7^2}\times \sqrt{6}$
$=\sqrt{4^2\times 12}\times \sqrt{7^2\times 6}$
$=\sqrt{4^2\times 12\times 7^2\times 6}$
$=\sqrt{4^2\times 6\times 2\times 7^2\times 6}$
$=\sqrt{4^2\times 6^2\times 7^2\times 2}$
$=\sqrt{4^2}\sqrt{6^2}\sqrt{7^2}\sqrt{2}=\mathrm{4.6.7.}\sqrt{2}=168\sqrt{2}$
(iii) $\sqrt[3]{332}\sqrt[3]{3250}=\sqrt[3]{38\times 4}\sqrt[3]{3125\times 2}$
$=\sqrt[3]{38}\sqrt[3]{34}\sqrt[3]{3125}\sqrt[3]{32}=\sqrt[3]{32^3}\sqrt[3]{34}\sqrt[3]{35^3}\sqrt[3]{32}$
$=2.\sqrt[3]{34}.5.\sqrt[3]{32}=\mathrm{2.5.}\sqrt[3]{34}.3\sqrt{2}$
$=10.\sqrt[3]{34\times 2}=10.\sqrt[3]{38}=10.\sqrt[3]{32^3}$
$=10.2=20$
(iv) $5.\sqrt{32}\times 2.\sqrt[3]{381}$
$=5.\sqrt{16\times 2}\times 2.\sqrt[3]{327\times 3}$
$=5.\sqrt{16}.\sqrt{2}.2.\sqrt[3]{327}.\sqrt[3]{33}$
$=5.\sqrt{4^2}.\sqrt{2}.2.\sqrt[3]{33^3}.\sqrt[3]{33^3}.\sqrt[3]{33}$
$=\mathrm{5.4.}\sqrt{2}.2.3.\sqrt[3]{33}$
$=120.\sqrt[2]{22}.\sqrt[3]{33}=120.\sqrt[2\times 3]{2\times 32^3}.\sqrt[3\times 2]{3\times 23^2}$
$=120.\sqrt[6]{68}.\sqrt[6]{69}=120.\sqrt[6]{68\times 9}=120.\sqrt[6]{672}$