Question

Simplify each of the following expressions:
(i) $(3+\sqrt{3})(2+\sqrt{2})$
(ii) $(3+\sqrt{3})(3-\sqrt{3})$ .
(iii) ${(\sqrt{5}+\sqrt{2})}^2$
(iv) $(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})$ .

[ 4 Marks ]

Answer 1

Step: Simplify the expression

(i) $(3+\sqrt{3})(2+\sqrt{2})$

Using $(a+\sqrt{b})(c+\sqrt{d})=ac+a\sqrt{d}+c\sqrt{b}+\sqrt{bd}$

$=(3\times 2)+(3\times \sqrt{2})+(2\times \sqrt{3})+(\sqrt{2}\times \sqrt{3})$

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

Hence, $(3+\sqrt{3})(2+\sqrt{2})=6+3\sqrt{2}+2\sqrt{3}+\sqrt{6}$

[1 Mark ]

Step: Simplify the expression

(ii) $(3+\sqrt{3})(3-\sqrt{3})$

Using, $(a+\sqrt{b})(a-\sqrt{b})=a^2-(\sqrt{b}{)}^2$

$=3^2-\left({\sqrt{3}}^2\right)$

$=9-3$

$=6$

Hence, $(3+\sqrt{3})(3-\sqrt{3})=6$

[ 1 Mark ]

(iii) ${(\sqrt{5}+\sqrt{2})}^2$ .

${(\sqrt{5}+\sqrt{2})}^2=(\sqrt{5}{)}^2+(\sqrt{2}{)}^2+2\times \sqrt{5}\times \sqrt{2}[\because (a+b{)}^2=a^2+b^2+2ab]=5+2+2\times \sqrt{5}\times \sqrt{2}=7+2\sqrt{10}$

[ 1 Mark ]

(iv) $(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})$ .

$(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})$$=(\sqrt{5}{)}^2-(\sqrt{2}{)}^2[\because (a+b\left)\right(a-b)=a^2-b^2]$
= 5-2 = 3

[ 1 Mark ]