If $\frac{7+\sqrt{5}}{7-\sqrt{5}}=a+b \sqrt{5}$, find a and b.

Given :

The given expression is $\frac{7+\sqrt{5}}{7-\sqrt{5}}=a+b \sqrt{5}$

To do :

We have to find the values of a and b.

Solution :

$\frac{7+\sqrt{5}}{7-\sqrt{5}}$

To factorize the denominator, multiply and divide by $7+\sqrt{5}$

LHS

$\frac{7+\sqrt{5}}{7-\sqrt{5}} = \frac{(7+\sqrt{5}) \times (7+ \sqrt{5})}{(7-\sqrt{5}) \times (7 + \sqrt{5})}$

                $=\frac{(7+\sqrt{5})^2}{7^2-\sqrt{5}^2} $

                $ = \frac{7^2 + \sqrt{5}^2 + 2 \times 7\sqrt{5}}{49-5} $

                $ = \frac{49 + 5 + 14\sqrt{5}}{44}$

                $ = \frac{54 + 14\sqrt{5}}{44}$

                 $ = \frac{54}{44} + \frac{14 \sqrt{5}}{44}$

                 $ = \frac{27}{22} + \frac{7\sqrt{5}}{22}$

RHS

                $ a + b\sqrt{5}$

On comparing, 

 $ a + b\sqrt{5} = \frac{27}{22} + \frac{7\sqrt{5}}{22}$

$a = \frac{27}{22}$

$b = \frac{7}{22}$.

Therefore, the values of a and b are  $\frac{27}{22},  \frac{7}{22}$.

Tutorialspoint

Updated on: 10-Oct-2022

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