Question

find the values of a and b

Answer 1

For rationalising the denominator we try and multiply the numerator and denominator with a number that would convert the denominator into a rational number . This we get a form of the same fraction that has a rational denominator and it also eases our calculation

Answer 2

Hey friend,
Here is the answer you were looking for:
$\frac{2 + \sqrt{3} }{2 - \sqrt{3} } = a + b \sqrt{3}$

On rationalizing the denominator we get,

$= \frac{2 + \sqrt{3} }{2 - \sqrt{3} } \times \frac{2 + \sqrt{3} }{2 + \sqrt{3} }$

Using the identities:

${(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab \\ (a + b)(a - b) = {a}^{2} - {b}^{2}$

$= \frac{ {(2)}^{2} + {( \sqrt{3} )}^{2} + 2(2)( \sqrt{3} )}{ {(2)}^{2} - {( \sqrt{3} )}^{2} } \\ \\ = \frac{4 + 3 + 4 \sqrt{3} }{4 - 3} \\ \\ = 7 + 4 \sqrt{3} \\ \\ 7 + 4 \sqrt{3} = a + b \sqrt{3} \\ \\ a = 7 \: : \: b = 4$

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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