Question

Find sum of the following surds:
$(\sqrt{12}+\sqrt{20}),(3\sqrt{3}+2\sqrt{5}),(\sqrt{45}-\sqrt{90})$.

Answer 1

The sum of the given surds can be determined as follows:

$(\sqrt{12}+\sqrt{20})+(3\sqrt{3}+2\sqrt{5})+(\sqrt{45}-\sqrt{90})=(\sqrt{4\times 3}+\sqrt{4\times 5})+(3\sqrt{3}+2\sqrt{5})+(\sqrt{9\times 5}-\sqrt{9\times 10})$

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

$=\sqrt{3}(2+3)+\sqrt{5}(2+2+3)-3\sqrt{10}=5\sqrt{3}+7\sqrt{5}-3\sqrt{10}$

Hence, $(\sqrt{12}+\sqrt{20})+(3\sqrt{3}+2\sqrt{5})+(\sqrt{45}-\sqrt{90})=5\sqrt{3}+7\sqrt{5}-3\sqrt{10}$.