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

Answer 1

(i) $(11+\sqrt{11})(11-\sqrt{11})=(11{)}^2-(\sqrt{11}{)}^2\{\because (a+b\left)\right(a-b)=a^2-b^2\}=121-11=110$

(ii) $(5+\sqrt{7})(5-\sqrt{7})=(5{)}^2-(\sqrt{7}{)}^2\{\because (a+b\left)\right(a-b)=a^2-b^2\}=25-7=18$

(iii) $(\sqrt{8}-\sqrt{2})(\sqrt{8}+\sqrt{2})=(\sqrt{8}{)}^2-(\sqrt{2}{)}^2\{\because (a+b\left)\right(a-b)=a^2-b^2\}=8-2=6$

(iv) $(3+\sqrt{3})(3-\sqrt{3})=(3{)}^2-(\sqrt{3}{)}^2\{\because (a+b\left)\right(a-b)=a^2-b^2\}=9-3=6$

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