Question

If a and b are rational, find a and b
5+2√3 / 7+4√3 = a+b√3

Answer 1

Hey mate ....☺

Here's ur ans. ----

5+ 2√3 / 7+4√3 × [7-4√3 /7-4√3]

5+2√3(7-4√3) ÷ { 7^2 - (4√3)^2}. 【 (a-b ) (a+b ) = a^2 - b^2 】

{35 + 14√3 - 20√3 - 24 } ÷ {49 - 48}

11- 6√3 ÷ 1
11- 6√3 = a-b√3

On comparing

a= 11 and
b = 6

✌✌Hope it helps u

Answer 2

Hey friend,
Here is the answer you were looking for:
$\frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } = a + b \sqrt{3} \\ \\ on \: rationalizing \: we \: get \\ \\ = \frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } \times \frac{7 - 4 \sqrt{3} }{7 - 4 \sqrt{3} } \\ \\ using \: the \: identity \\ (a + b)(a - b) = {a}^{2} - {b}^{2} \\ \\ = \frac{5 \times 7 - 5 \times 4 \sqrt{3} + 2 \sqrt{3} \times 7 - 2 \sqrt{3} \times 4 \sqrt{3} }{ {(7)}^{2} - {(4 \sqrt{3}) }^{2} } \\ \\ = \frac{35 - 20 \sqrt{3} + 14 \sqrt{3} - 8 \times 3}{49 - 48} \\ \\ = 35 - 24 - 20 \sqrt{3} + 14 \sqrt{3} \\ \\ 11 - 6 \sqrt{3} = a + b \sqrt{3} \\ \\ a = 11 \\ \\ b = - 6$

Hope this helps!!!!

@Mahak24

Thanks....
☺☺