Question

If $\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=a+b\sqrt{3}$ , then the value of $a$ and $b$ is

Answer 1

Step 1: Given fraction.

The given fraction is $\frac{5+2\sqrt{3}}{7+4\sqrt{3}}$, which can be written in the form $a+b\sqrt{3}$.

Step 2: Rationalize the denominator.

$\frac{5+2\sqrt{3}}{7+4\sqrt{3}}\times \frac{7-4\sqrt{3}}{7-4\sqrt{3}}$

$=\frac{5(7-4\sqrt{3})+2\sqrt{3}(7-4\sqrt{3})}{(7+4\sqrt{3})(7-4\sqrt{3})}$

$=\frac{35-20\sqrt{3}+14\sqrt{3}-8\left(3\right)}{7^2-{\left(4\sqrt{3}\right)}^2}\mathbf{\left[}\mathbf{\right(}\mathbf{a}\mathbf{+}\mathbf{b}\mathbf{\left)}\mathbf{\right(}\mathbf{a}\mathbf{-}\mathbf{b}\mathbf{)}\mathbf{=}{\mathbf{a}}^{\mathbf{2}}\mathbf{-}{\mathbf{b}}^{\mathbf{2}}]$

$=\frac{35-24+(-20+14)\sqrt{3}}{49-16\left(3\right)}$

$=\frac{11-6\sqrt{3}}{49-48}$

$=11-6\sqrt{3}$

Hence, the values of $\mathit{a}$ and $\mathit{b}$ are $\mathbf{11}$ and $\mathbf{-}\mathbf{6}$ respectively.