Question

Find the values of $a$ and $b$

$if\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=a+b\sqrt{3}$

• A. a= 11, b = -6
• B. a = 9, b = -6
• C. a= -11, b = 6
• D. a= -9, b = 6

Answer 1

The correct option is A

a= 11, b = -6

$\frac{5+2\sqrt{3}}{7+4\sqrt{3}}$ can be simplified by rationalising the denominator $(7+4\sqrt{3})$. We can rationalize the given denominator by multiplying the given number $(7+4\sqrt{3})$ with its rationalizing factor $(7-4\sqrt{3})$. Now if we multiply the number $(7-4\sqrt{3})$ in the denominator, we also have to multiply the same number in the numerator.

On multiplying $(7-4\sqrt{3})$ in the numerator and denominator, we get

$\frac{(5+2\sqrt{3})(7-4\sqrt{3})}{(7+4\sqrt{3})(7-4\sqrt{3})}$=$\frac{35+14\sqrt{3}-20\sqrt{3}-8\left(\sqrt{3}\sqrt{3}\right)}{\left(7\right)\left(7\right)-(4\times 4)(\sqrt{3}\times \sqrt{3})}=11-6\sqrt{3}$

On comparing the answer found, with $a+b\sqrt{3}$, we get that $a=11,b=-6$