Question

If $a$ and $b$ is rational no. and $\frac{3+\sqrt{7}}{3-\sqrt{7}}=a+b\sqrt{7}$, find the value of $a$ and $b$.

Answer 1

On simplifying,

$\frac{3+\sqrt{7}}{3-\sqrt{7}}=a+b\sqrt{7}$

$\Rightarrow \frac{3+\sqrt{7}}{3-\sqrt{7}}\times \frac{3+\sqrt{7}}{3+\sqrt{7}}=a+b\sqrt{7}$

$\Rightarrow \frac{(3+\sqrt{7}{)}^2}{3^2-(\sqrt{7}{)}^2}=a+b\sqrt{7}$

$\Rightarrow \frac{9+7+6\sqrt{7}}{2}=a+\sqrt{7}$

$\Rightarrow \frac{16+6\sqrt{7}}{2}=a+\sqrt{7}$

$\Rightarrow \frac{2(8+3\sqrt{7})}{2}=a+\sqrt{7}$

$\Rightarrow 8+3\sqrt{7}=a+b\sqrt{7}$.

Therefore, $a=8,b=3$