Simplify the following expressions:
Question:

Simplify the following expressions:

(i) $(11+\sqrt{11})(11-\sqrt{11})$

(ii) $(5+\sqrt{7})(5-\sqrt{7})$

(iii) $(\sqrt{8}-\sqrt{2})(\sqrt{8}+\sqrt{2})$

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

(v) $(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})$

Solution:

(i) $(11+\sqrt{11})(11-\sqrt{11})$

As we know, $(a+b)(a-b)=\left(a^{2}-b^{2}\right)$

So, $11^{2}-11$

$121-11=110$

(ii) $(5+\sqrt{7})(5-\sqrt{7})$

As we know, $(a+b)(a-b)=\left(a^{2}-b^{2}\right)$

So, $5^{2}-7$

$25-7=18$

(iii) $(\sqrt{8}-\sqrt{2})(\sqrt{8}+\sqrt{2})$

As we know, $(a+b)(a-b)=\left(a^{2}-b^{2}\right)$

$\sqrt{8 \times 8}-\sqrt{2 \times 2}=8-2$

$=6$

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

As we know, $(a+b)(a-b)=\left(a^{2}-b^{2}\right)$

$=9-\sqrt{3 \times 3}$

$=6$

(v) $(\sqrt{5}-\sqrt{2})(\sqrt{5}+\sqrt{2})$

As we know, $(a+b)(a-b)=\left(a^{2}-b^{2}\right)$

$=\sqrt{5 \times 5}-\sqrt{2 \times 2}$

$=5-2$

$=3$