Simplify

Question:

Simplify

(i) $3 \sqrt{45}-\sqrt{125}+\sqrt{200}-\sqrt{50}$

(ii) $\frac{2 \sqrt{30}}{\sqrt{6}}-\frac{3 \sqrt{140}}{\sqrt{28}}+\frac{\sqrt{55}}{\sqrt{99}}$

(iii) $\sqrt{72}+\sqrt{800}-\sqrt{18}$

Solution:

(i) $3 \sqrt{45}-\sqrt{125}+\sqrt{200}-\sqrt{50}$

$=3 \sqrt{9 \times 5}-\sqrt{25 \times 5}+\sqrt{100 \times 2}-\sqrt{25 \times 2}$

$=3 \times 3 \sqrt{5}-5 \sqrt{5}+10 \sqrt{2}-5 \sqrt{2}$

$=9 \sqrt{5}-5 \sqrt{5}+5 \sqrt{2}$

$=4 \sqrt{5}+5 \sqrt{2}$

(ii) $\frac{2 \sqrt{30}}{\sqrt{6}}-\frac{3 \sqrt{140}}{\sqrt{28}}+\frac{\sqrt{55}}{\sqrt{99}}$

$=\frac{2 \sqrt{6 \times 5}}{\sqrt{6}}-\frac{3 \sqrt{28 \times 5}}{\sqrt{28}}+\frac{\sqrt{5 \times 11}}{\sqrt{9 \times 11}}$

$=\frac{2 \sqrt{6} \times \sqrt{5}}{\sqrt{6}}-\frac{3 \sqrt{28} \times \sqrt{5}}{\sqrt{28}}+\frac{\sqrt{5} \times \sqrt{11}}{\sqrt{9} \times \sqrt{11}}$

$=2 \sqrt{5}-3 \sqrt{5}+\frac{\sqrt{5}}{3}$

$=-\sqrt{5}+\frac{\sqrt{5}}{3}$

$=\frac{-3 \sqrt{5}+\sqrt{5}}{3}$

$=\frac{-2 \sqrt{5}}{3}$

(iii) $\sqrt{72}+\sqrt{800}-\sqrt{18}$

$=\sqrt{36 \times 2}+\sqrt{400 \times 2}-\sqrt{9 \times 2}$

$=6 \sqrt{2}+20 \sqrt{2}-3 \sqrt{2}$

$=23 \sqrt{2}$