Question

Rationalize the denominator of the following: 1] 1/ 3 + √ 5 - 2√2

Answer 1

Dear Student,

$$\begin{gathered} Asyourquestioniswrittenas \\ \frac{1}{3+\sqrt{5}-2\sqrt{2}} \\ Onrationalizing \\ \Rightarrow \frac{{\displaystyle 1}}{{\displaystyle 3+\left(\sqrt{5}-2\sqrt{2}\right)}}\times \frac{3-\left(\sqrt{5}-2\sqrt{2}\right)}{3-\left(\sqrt{5}-2\sqrt{2}\right)} \\ \Rightarrow \frac{3-\left(\sqrt{5}-2\sqrt{2}\right)}{{\displaystyle {\left(3\right)}^3}-{\left(\sqrt{5}-2\sqrt{2}\right)}^2}\left[\left(a+b\right)\left(a-b\right)=a^2-b^2\right] \\ \Rightarrow \frac{{\displaystyle 3-\left(\sqrt{5}-2\sqrt{2}\right)}}{{\displaystyle 9-\left(5+8-2\sqrt{2}\sqrt{5}\right)}}\left[{\left(a-b\right)}^2=a^2+b^2-2ab\right] \\ \Rightarrow \frac{{\displaystyle 3-\left(\sqrt{5}-2\sqrt{2}\right)}}{{\displaystyle 9-\left(13-2\sqrt{2}\sqrt{5}\right)}} \\ Againwe\mathrm{Rationalize}\mathrm{the}denominator \\ \Rightarrow \frac{{\displaystyle 3-\left(\sqrt{5}-2\sqrt{2}\right)}}{{\displaystyle 2\sqrt{10}-4}}\times \frac{2\sqrt{10}+4}{2\sqrt{10}+4} \\ \Rightarrow \frac{{\displaystyle \left[3-\sqrt{5}+2\sqrt{2}\right]\left[2\sqrt{10}+4\right]}}{{\displaystyle {\left(2\sqrt{10}\right)}^2-4^2}} \\ \Rightarrow \frac{6\sqrt{10}-2\sqrt{50}+4\sqrt{20}+12-4\sqrt{5}+8\sqrt{2}}{40-16} \\ \Rightarrow \frac{{\displaystyle 6\sqrt{10}-2\sqrt{2\times 5\times 5}+4\sqrt{2\times 2\times 5}+12-4\sqrt{5}+8\sqrt{2}}}{{\displaystyle 40-16}} \\ \Rightarrow \frac{6\sqrt{10}-10\sqrt{2}+8\sqrt{5}+12-4\sqrt{5}+8\sqrt{2}}{24} \\ \Rightarrow \frac{6\sqrt{10}-2\sqrt{2}+4\sqrt{5}+12}{24} \\ \Rightarrow \frac{\mathbf{3}\sqrt{\mathbf{10}}\mathbf{-}\sqrt{\mathbf{2}}\mathbf{+}\mathbf{2}\sqrt{\mathbf{5}}\mathbf{+}\mathbf{6}}{\mathbf{12}} \end{gathered}$$
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