Question

Simplify

(i) $3\sqrt{45}-\sqrt{125}+\sqrt{200}-\sqrt{50}$
(ii) $2\frac{\sqrt{30}}{\sqrt{6}}-3\frac{\sqrt{140}}{\sqrt{28}}+\frac{\sqrt{55}}{\sqrt{99}}$
(iii) $\sqrt{72}+\sqrt{800}-\sqrt{18}$

Answer 1

(i) Given $3\sqrt{45}-\sqrt{125}+\sqrt{200}-\sqrt{50}$
$=3\sqrt{9\times 5}-\sqrt{25\times 5}+\sqrt{100\times 2}-\sqrt{25\times 2}$
$=(3\times 3)\sqrt{5}-5\sqrt{5}+10\sqrt{2}-5\sqrt{2}$
$=9\sqrt{5}-5\sqrt{5}+5\sqrt{2}$
$=4\sqrt{5}+5\sqrt{2}$
Therefore, the value of the given expression is $4\sqrt{5}+5\sqrt{2}$
(ii) Given $2\frac{\sqrt{30}}{\sqrt{6}}-3\frac{\sqrt{140}}{\sqrt{28}}+\frac{\sqrt{55}}{\sqrt{99}}$
$=2\frac{\sqrt{6}\times \sqrt{5}}{\sqrt{6}}-3\frac{\sqrt{28}\times \sqrt{5}}{\sqrt{28}}+\frac{\sqrt{5}\times \sqrt{11}}{\sqrt{9}\times \sqrt{11}}$
$=2\times \sqrt{5}-3\times \sqrt{5}+\frac{\sqrt{5}}{3}$
$=-\sqrt{5}+\frac{\sqrt{5}}{3}$
$=\frac{(-3+1)\times \sqrt{5}}{3}$
$=\frac{-2\times \sqrt{5}}{3}$
Hence, the value of the given expression is $\frac{-2\times \sqrt{5}}{3}$
(iii) Given $\sqrt{72}+\sqrt{800}-\sqrt{18}$
$=\sqrt{36}\times \sqrt{2}+\sqrt{400}\times \sqrt{2}-\sqrt{9}\times \sqrt{2}$
$=6\sqrt{2}+20\times \sqrt{2}-3\sqrt{2}$
$=23\sqrt{2}$
Hence, the value of the given expression is $23\sqrt{2}$