Simplify

Question:

Simplify

(i) $(3-\sqrt{11})(3+\sqrt{11})$

(ii) $(-3+\sqrt{5})(-3-\sqrt{5})$

(iii) $(3-\sqrt{3})^{2}$

(iv) $(\sqrt{5}-\sqrt{3})^{2}$

(v) $(5+\sqrt{7})(2+\sqrt{5})$

(vi) $(\sqrt{5}-\sqrt{2})(\sqrt{2}-\sqrt{3})$

 

Solution:

(i) $(3-\sqrt{11})(3+\sqrt{11})$

$=3^{2}-(\sqrt{11})^{2} \quad\left[(a-b)(a+b)=a^{2}-b^{2}\right]$

$=9-11$

$=-2$

(ii) $(-3+\sqrt{5})(-3-\sqrt{5})$

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

$=9-5$

$=4$

(iii) $(3-\sqrt{3})^{2}$

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

$=9+3-6 \sqrt{3}$

$=12-6 \sqrt{3}$

(iv) $(\sqrt{5}-\sqrt{3})^{2}$

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

$=5+3-2 \sqrt{15}=8-2 \sqrt{15}$

$=\sqrt{5} \times \sqrt{2}-\sqrt{5} \times \sqrt{3}-\sqrt{2} \times \sqrt{2}+\sqrt{2} \times \sqrt{3}$

$=\sqrt{10}-\sqrt{15}-2+\sqrt{6}$

(v) $(5+\sqrt{7})(2+\sqrt{5})$

$(5+\sqrt{7})(2+\sqrt{5})$

$=5 \times 2+5 \times \sqrt{5}+\sqrt{7} \times 2+\sqrt{7} \times \sqrt{5}$

$=10+5 \sqrt{5}+2 \sqrt{7}+\sqrt{35}$

(vi) $(\sqrt{5}-\sqrt{2})(\sqrt{2}-\sqrt{3})$

$(\sqrt{5}-\sqrt{2})(\sqrt{2}-\sqrt{3})$

$=\sqrt{5} \times \sqrt{2}-\sqrt{5} \times \sqrt{3}-\sqrt{2} \times \sqrt{2}+\sqrt{2} \times \sqrt{3}$

$=\sqrt{10}-\sqrt{15}-2+\sqrt{6}$

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