Question

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Q.1. Find the value of a and b if $\frac{5+\sqrt{3}}{7-4\sqrt{3}}=94a+3\sqrt{3}b$.

Answer 1

$$\begin{gathered} \frac{5+\sqrt{3}}{7-4\sqrt{3}}\times \frac{7+4\sqrt{3}}{7+4\sqrt{3}} \\ =\frac{{\displaystyle \left(5+\sqrt{3}\right)\left(7+4\sqrt{3}\right)}}{{\displaystyle {\left(7\right)}^2-{\left(4\sqrt{3}\right)}^2}} \\ =\frac{{\displaystyle \left(35+7\sqrt{3}+20\sqrt{3}+12\right)}}{{\displaystyle 49-48}} \\ =47+27\sqrt{3} \\ =94\left(\frac{1}{2}\right)+3\sqrt{3}\times 9 \\ Comparingwith94a+3\sqrt{3}b \\ \Rightarrow a=\frac{1}{2}andb=9 \end{gathered}$$