If $\dfrac{5 + 2\sqrt{3}}{7 + 4\sqrt{3}} = a - b\sqrt{3}$, find the values of 'a' and 'b'.

Given,

Equation : $\dfrac{5 + 2\sqrt{3}}{7 + 4\sqrt{3}} = a - b\sqrt{3}$

Rationalizing the denominator of L.H.S. of the above equation :

$\Rightarrow \dfrac{5 + 2\sqrt{3}}{7 + 4\sqrt{3}} \times \dfrac{7 - 4\sqrt{3}}{7 - 4\sqrt{3}} \\[1em] \Rightarrow \dfrac{(5 + 2\sqrt{3}) \times (7 - 4\sqrt{3})}{(7)^2 - (4\sqrt{3})^2} \\[1em] \Rightarrow \dfrac{35 - 20\sqrt{3} + 14\sqrt{3} - 8 \times 3}{49 - 48} \\[1em] \Rightarrow \dfrac{35 - 6\sqrt{3} - 24}{1} \\[1em] \Rightarrow 11 - 6\sqrt{3} \\[1em]$

Comparing, $11 - 6\sqrt{3} \text{ with } a - b\sqrt{3}$, we get :

a = 11 and b = 6.

Hence, a = 11 and b = 6.