Question

Simplify the following:
$\left(i\right)\sqrt{45}-3\sqrt{20}+4\sqrt{5}\left(ii\right)\frac{\sqrt{24}}{8}+\frac{\sqrt{54}}{9}\left(iii\right)4\sqrt{12}\times 7\sqrt{6}$

Answer 1

$\left(i\right)\sqrt{45}-3\sqrt{20}+4\sqrt{5}=\sqrt{3\times 3\times 5}-3\sqrt{2\times 2\times 5}+4\sqrt{5}[\because 45=3\times 3\times 5,20=2\times 2\times 5]=3\sqrt{5}-3\times 2\sqrt{5}+4\sqrt{5}=3\sqrt{5}-6\sqrt{5}+4\sqrt{5}=\sqrt{5}$

$\left(ii\right)\frac{\sqrt{24}}{8}+\frac{\sqrt{54}}{9}=\frac{\sqrt{2\times 2\times 2\times 3}}{8}+\frac{\sqrt{3\times 3\times 3\times 2}}{9}[\because 24=2\times 2\times 2\times 3,54=2\times 3\times 3\times 3]=\frac{2\sqrt{6}}{8}+\frac{3\sqrt{6}}{9}=\frac{\sqrt{6}}{4}+\frac{\sqrt{6}}{3}=\frac{3\sqrt{6}+4\sqrt{6}}{12}=\frac{7\sqrt{6}}{12}$

$\left(iii\right)4\sqrt{12}\times 7\sqrt{6}=4\sqrt{2\times 2\times 3}\times 7\sqrt{2\times 3}[\because 12=2\times 2\times 3,6=2\times 3]$
$=8\sqrt{3}\times 7\sqrt{2\times 3}$