Question

find the value of a and b

Answer 1

Hey friend,
Here is the answer you were looking for:
$\frac{3 + \sqrt{7} }{3 - \sqrt{7} } = a + b \sqrt{7}$

On rationalizing the denominator we get,

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

Using the identities:
${(a + b) }^{2} = {a}^{2} + {b}^{2} + 2ab \\ (a + b)(a - b) = {a}^{2} - {b}^{2}$

Putting a = 3
and b = √7

$= \frac{ {(3)}^{2} + {( \sqrt{7} )}^{2} + 2(3)( \sqrt{7}) }{(3)^{2} - ( \sqrt{7})^{2} } \\ \\ = \frac{9 + 7 + 6 \sqrt{7} }{9 - 7} \\ \\ = \frac{16 + 6 \sqrt{7} }{2} \\ \\ = \frac{2(8 + 3 \sqrt{7}) }{2} \\ \\ = 8 + 3 \sqrt{7} \\ \\ 8 + 3\sqrt{7} = a + b \sqrt{7} \\ \\ a = 8 \: : b = 3$

Hope this helps!!

If you have any doubt regarding to my answer, feel free to ask in the comment section or inbox me if needed.

@Mahak24

Thanks...
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