Question

Simplify: $(3\sqrt{27}+5)(9\sqrt{3}+7)$.

Answer 1

The given surds can be simplified as follows:

$(3\sqrt{27}+5)(9\sqrt{3}+7)=3\sqrt{27}(9\sqrt{3}+7)+5(9\sqrt{3}+7)=(3\times 9\sqrt{27\times 3})+(3\times 7\sqrt{27})-(5\times 9\sqrt{3})+(5\times 7)$

$=27\sqrt{81}+21\sqrt{27}+45\sqrt{3}+35=27\sqrt{9\times 9}+21\sqrt{9\times 3}+45\sqrt{3}+35=27\sqrt{9^2}+21\sqrt{3^2\times 3}+45\sqrt{3}+35$

$=(27\times 9)+(21\times 3\sqrt{3})+45\sqrt{3}+35$

$=243+63\sqrt{3}+45\sqrt{3}+35=278+108\sqrt{3}$ .

Hence, $(3\sqrt{27}+5)(9\sqrt{3}+7)=278+108\sqrt{3}$ .