Question

Find the values of a and b in each of the following.
$\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=a-b\sqrt{3}$

Answer 1

We have :
$\frac{5+2\sqrt{3}}{7+4\sqrt{3}}$

Now, rationalising the denominator of the given number by multiplying both the numerator and denominator with $7-4\sqrt{3}$, we get:
$$\begin{gathered} \frac{5+2\sqrt{3}}{7+4\sqrt{3}}=\frac{5+2\sqrt{3}}{7+4\sqrt{3}}\times \frac{7-4\sqrt{3}}{7-4\sqrt{3}} \\ =\frac{35+14\sqrt{3}-20\sqrt{3}-24}{{\left(7\right)}^2-{\left(4\sqrt{3}\right)}^2} \\ =\frac{11-6\sqrt{3}}{49-48} \\ =11-6\sqrt{3} \end{gathered}$$

∴ $11-6\sqrt{3}$ = a $-$ b$\sqrt{3}$
So, on comparing the LHS and the RHS, we get:
a = 11 and b = 6