Question

5+2√3÷7+4√3=a+b√3 find value of a and b

Answer 1

Hi...✌

Here is your answer...☺

On rationalising LHS
LHS =

$\frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } \times \frac{7 - 4 \sqrt{3} }{7 - 4 \sqrt{3} } \\ \\ = \frac{35 - 20 \sqrt{3} + 14 \sqrt{3} - 24 }{ {7}^{2} - ( {4 \sqrt{3} })^{2} } \\ \\ = \frac{11 - 6 \sqrt{3} }{49 - 48} = 11 - 6 \sqrt{3}$

Now
By comparing LHS and RHS

11-6√3 = a+b√3

We get,

a = 11 , b = -6

Thank you ⭐

Answer 2

Hope u like my process
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$= > \bf\frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } \\ \\ = > \frac{(5 + 2 \sqrt{3}) }{(7 + 4 \sqrt{3} )} \times \frac{(7 - 4 \sqrt{3}) }{(7 - 4 \sqrt{3}) } \\ \\ = > \frac{(5 + 2 \sqrt{3} )(7 - 4 \sqrt{3} )}{ {7}^{2} - {(4 \sqrt{3}) }^{2} } \\ \\ = > \frac{(7 \times 5) + (7 \times 2 \sqrt{3}) - (5 \times 4 \sqrt{3} ) - (4 \sqrt{3} \times 2 \sqrt{3}) }{49 - 48} \\ \\ = > 35 + 14 \sqrt{3} - 20 \sqrt{3 } - 24 \\ \\ = > \bf 11 - 6 \sqrt{3}$
Now..

$= > \boxed{\bf \: a + b \sqrt{3} = \it \frac{5 + 2 \sqrt{3} }{7 + 4 \sqrt{3} } = \bf \underline{11 - 6 \sqrt{3} } }$
thus..

=> a = 11 ____(answer)

=> b =-6 ____(answer)
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Hope this is ur required answer

proud to help u