Question

Simplify
(i) $10\sqrt{2}-2\sqrt{2}+4\sqrt{32}$
(ii) $\sqrt{48}-3\sqrt{72}-\sqrt{27}+5\sqrt{18}$
(iii) $\sqrt[3]{316}+8\sqrt[3]{354}-\sqrt[3]{3128}$.

Answer 1

(i) $10\sqrt{2}-2\sqrt{2}+4\sqrt{32}$
$=10\sqrt{2}-2\sqrt{2}+4\sqrt{16\times 2}$
$=10\sqrt{2}-2\sqrt{2}+4\times 4\times \sqrt{2}$
$=(10-2+16)\sqrt{2}=24\sqrt{2}$
(ii) $\sqrt{48}-3\sqrt{72}-\sqrt{27}+5\sqrt{18}$
$=\sqrt{16\times 3}-3\sqrt{36\times 2}-\sqrt{9\times 3}+5\sqrt{9\times 2}$
$=\sqrt{16}\sqrt{3}-3\sqrt{36}\sqrt{2}=\sqrt{9}\sqrt{3}+5\sqrt{9}\sqrt{2}$
$=4\sqrt{3}-18\sqrt{2}-3\sqrt{3}+15\sqrt{2}$
$=(-18+15)\sqrt{2}+(4-3)\sqrt{3}=-3\sqrt{2}+\sqrt{3}$
(iii) $\sqrt[3]{316}+8\sqrt[3]{354}-\sqrt[3]{3128}$
$=\sqrt[3]{38\times 2}+8\sqrt[3]{327\times 2}-\sqrt[3]{364\times 2}$
$=\sqrt[3]{38}\sqrt[3]{32}+8\sqrt[3]{327}\sqrt[3]{32}-\sqrt[3]{364}\sqrt[3]{32}$
$=2\sqrt[3]{32}+8\times 3\times \sqrt[3]{32}-4\sqrt[3]{32}$
$=2\sqrt[3]{32}+24\sqrt[3]{32}-4\sqrt[3]{32}$
$=(2+24-4)\sqrt[3]{32}=22\sqrt[3]{32}$