Question

find the value of a and b:

Answer 1

We have to rationalize the denominator by multiplying with its conjugate irrational expression.  So we multiply the numerator and denominator with  7 - √3 on the LHS.

$\frac{5 + 2 \sqrt3}{7+\sqrt3}=a-\sqrt3 b\\\\\frac{(5 + 2 \sqrt3)(7-\sqrt3)}{(7+\sqrt3)(7-\sqrt3)}=a-\sqrt3 b\\\\\frac{35-2*3+9\sqrt3}{7^2-3}=a-b\sqrt3\\\\a=\frac{29}{46}\\\\b=\frac{-9}{46}$

That is the answer.