Question

Simplify
(i) $\left(3-\sqrt{11}\right)\left(3+\sqrt{11}\right)$
(ii) $\left(-3+\sqrt{5}\right)\left(-3-\sqrt{5}\right)$
(iii) ${\left(3-\sqrt{3}\right)}^2$
(iv) ${\left(\sqrt{5}-\sqrt{3}\right)}^2$
(v) $\left(5+\sqrt{7}\right)\left(2+\sqrt{5}\right)$
(vi) $\left(\sqrt{5}-\sqrt{2}\right)\left(\sqrt{2}-\sqrt{3}\right)$

Answer 1

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

$$\begin{gathered} =3^2-{\left(\sqrt{11}\right)}^2[\left(a-b\right)\left(a+b\right)=a^2-b^2] \\ =9-11 \\ =-2 \end{gathered}$$

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

$$\begin{gathered} ={\left(-3\right)}^2-{\left(\sqrt{5}\right)}^2[\left(a+b\right)\left(a-b\right)=a^2-b^2] \\ =9-5 \\ =4 \end{gathered}$$

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

$$\begin{gathered} =3^2+{\left(\sqrt{3}\right)}^2-2\times 3\times \sqrt{3}[{\left(a-b\right)}^2=a^2+b^2-2ab] \\ =9+3-6\sqrt{3} \\ =12-6\sqrt{3} \end{gathered}$$

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

$$\begin{gathered} ={\left(\sqrt{5}\right)}^2+{\left(\sqrt{3}\right)}^2-2\times \sqrt{5}\sqrt{3}[{\left(a-b\right)}^2=a^2+b^2-2ab] \\ =5+3-2\sqrt{15}=8-2\sqrt{15} \end{gathered}$$
$$\begin{gathered} =\sqrt{5}\times \sqrt{2}-\sqrt{5}\times \sqrt{3}-\sqrt{2}\times \sqrt{2}+\sqrt{2}\times \sqrt{3} \\ =\sqrt{10}-\sqrt{15}-2+\sqrt{6} \end{gathered}$$

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

$$\begin{gathered} \left(5+\sqrt{7}\right)\left(2+\sqrt{5}\right) \\ =5\times 2+5\times \sqrt{5}+\sqrt{7}\times 2+\sqrt{7}\times \sqrt{5} \\ =10+5\sqrt{5}+2\sqrt{7}+\sqrt{35} \end{gathered}$$

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

$$\begin{gathered} \left(\sqrt{5}-\sqrt{2}\right)\left(\sqrt{2}-\sqrt{3}\right) \\ =\sqrt{5}\times \sqrt{2}-\sqrt{5}\times \sqrt{3}-\sqrt{2}\times \sqrt{2}+\sqrt{2}\times \sqrt{3} \\ =\sqrt{10}-\sqrt{15}-2+\sqrt{6} \end{gathered}$$