Question

$Q.11showthat5\sqrt{2}isnotrationalnumberQ.12Simplify3\sqrt[3]{16}+7\sqrt[3]{16}-4\sqrt[3]{54}Q.13simplify\frac{2+\sqrt{5}}{2-\sqrt{5}}+\frac{2-\sqrt{5}}{2+\sqrt{5}}$

Answer 1

Dear student
13
$$\begin{gathered} \frac{2+\sqrt{5}}{2-\sqrt{5}}+\frac{2-\sqrt{5}}{2+\sqrt{5}} \\ =\frac{{\displaystyle 2+\sqrt{5}}}{{\displaystyle 2-\sqrt{5}}}\times \frac{2+\sqrt{5}}{2+\sqrt{5}}+\frac{{\displaystyle 2-\sqrt{5}}}{{\displaystyle 2+\sqrt{5}}}\times \frac{2-\sqrt{5}}{2-\sqrt{5}} \\ =\frac{{\displaystyle {\left(2+\sqrt{5}\right)}^2}}{4-5}+\frac{{\displaystyle {\left(2-\sqrt{5}\right)}^2}}{4-5} \\ =-\left(4+5+4\sqrt{5}\right)-\left(4+5-4\sqrt{5}\right) \\ =-9-4\sqrt{5}-9+4\sqrt{5} \\ =-18 \\ \mathrm{Note}: \\ {\mathrm{a}}^2-{\mathrm{b}}^2=\left(\mathrm{a}+\mathrm{b}\right)\left(\mathrm{a}-\mathrm{b}\right) \\ {\left(\mathrm{a}\pm \mathrm{b}\right)}^2={\mathrm{a}}^2+{\mathrm{b}}^2\pm 2\mathrm{ab} \end{gathered}$$
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