Question

If both a and b are rational numbers, find the values of a and b in the following equalities.
3+?7/3-?7=a+b?7

Answer 1

$$\begin{gathered} \frac{3+\sqrt{7}}{3-\sqrt{7}}=\mathrm{a}+\mathrm{b}\sqrt{7} \\ \Rightarrow \frac{{\displaystyle 3+\sqrt{7}}}{{\displaystyle 3-\sqrt{7}}}\times \frac{{\displaystyle 3+\sqrt{7}}}{{\displaystyle 3+\sqrt{7}}}=\mathrm{a}+\mathrm{b}\sqrt{7} \\ \Rightarrow \frac{9+7+6\sqrt{7}}{9-7}=\mathrm{a}+\mathrm{b}\sqrt{7}\mathbf{}\left[\mathbf{as}\mathbf{,}\mathbf{}{\left(\mathbf{a}\mathbf{+}\mathbf{b}\right)}^{\mathbf{2}}\mathbf{}\mathbf{=}\mathbf{}{\mathbf{a}}^{\mathbf{2}}\mathbf{}\mathbf{+}\mathbf{}{\mathbf{b}}^{\mathbf{2}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{2}\mathbf{ab}\mathbf{}\mathbf{}\mathbf{and}\mathbf{}{\mathbf{a}}^{\mathbf{2}}\mathbf{}\mathbf{-}\mathbf{}{\mathbf{b}}^{\mathbf{2}}\mathbf{}\mathbf{=}\mathbf{}\left(\mathbf{a}\mathbf{+}\mathbf{b}\right)\left(\mathbf{a}\mathbf{-}\mathbf{b}\right)\right] \\ \Rightarrow \frac{16+6\sqrt{7}}{2}=\mathrm{a}+\mathrm{b}\sqrt{7} \\ \Rightarrow 8+3\sqrt{7}=\mathrm{a}+\mathrm{b}\sqrt{7} \\ \Rightarrow \mathrm{a}=8;\mathrm{b}=3 \end{gathered}$$