Question

If both a and b are the rational numbers, then find the value of a and b in:(5+2√3)/(7+4√3)= a+b√3.

Answer 1

we have to just rationalise
$\frac{(5 + 2 \sqrt{3}) }{(7 + 4 \sqrt{3} )}$
so here i go
$\frac{(5 + 2 \sqrt{3}) }{(7 + 4 \sqrt{3} )} \times \frac{(7 - 4 \sqrt{3}) }{(7 - 4 \sqrt{3}) } = \frac{35 - 20 \sqrt{3} + 14 \sqrt{3} - 24 }{ {7}^{2} - {(4 \sqrt{3}) }^{2} } \\ = \frac{11 - 6 \sqrt{3} }{49 - 48} = 11 - 6 \sqrt{3} \\ now \: we \: can \: see \: that \\ 11 - 6 \sqrt{3} = a + b \sqrt{3} \\ therefore \\ a = 11 \\ b = - 6$