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}) $

To do: 

We have to simplify the given expressions.

Solution:

We know that,

$(a+b)(a-b)=a^2-b^2$

$(a+b)^2=a^2+2ab+b^2$

$(a-b)^2=a^2-2ab+b^2$

$(a+b)(c+d)=a(c+d)+b(c+d)$

Therefore,

(i) $(3+\sqrt{3})(2+\sqrt{2})=3(2+\sqrt2)+\sqrt3(2+\sqrt2)$

$=3(2)+3\times \sqrt2+\sqrt3 \times2+\sqrt3 \times \sqrt2$

$=6+3\sqrt2+2\sqrt3+\sqrt{3\times2}$

$=6+3\sqrt2+2\sqrt3+\sqrt6$

(ii) $(3+\sqrt{3})(3-\sqrt{3})=(3)^2-(\sqrt3)^2$

$=9-3$

$=6$

(iii) $(\sqrt{5}+\sqrt{3})^{2}=(\sqrt{5})^{2}+(\sqrt{3})^{2}+2 \times \sqrt{5} \times \sqrt{3}$

$=5+3+2 \sqrt{5\times3}$

$=8+2 \sqrt{15}$

 (iv) $(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})=(\sqrt5)^2-(\sqrt2)^2$

$=5-2$

$=3$