Question

Find the value of A and B 5 first part 5 + 2 root 3 upon 7 + 4 root 3 is equals to a minus b root 3
Q. Find the value of a and b
(i). $\frac{5+2\sqrt{3}}{7+4\sqrt{3}}=a-b\sqrt{3}$

Answer 1

$$\begin{gathered} \frac{5+2\sqrt{3}}{7+4\sqrt{3}}=\mathrm{a}-\mathrm{b}\sqrt{3} \\ \Rightarrow \frac{{\displaystyle 5+2\sqrt{3}}}{{\displaystyle 7+4\sqrt{3}}}\times \frac{7-4\sqrt{3}}{7+4\sqrt{3}}=\mathrm{a}-\mathrm{b}\sqrt{3} \\ \Rightarrow \frac{35-20\sqrt{3}+14\sqrt{3}-24}{49-48}=\mathrm{a}-\mathrm{b}\sqrt{3}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\left[\mathbf{as}\mathbf{,}\mathbf{}\left(\mathbf{a}\mathbf{+}\mathbf{b}\right)\left(\mathbf{a}\mathbf{-}\mathbf{b}\right)\mathbf{}\mathbf{=}\mathbf{}{\mathbf{a}}^{\mathbf{2}}\mathbf{}\mathbf{-}\mathbf{}{\mathbf{b}}^{\mathbf{2}}\right] \\ \Rightarrow \frac{11-6\sqrt{3}}{1}=\mathrm{a}-\mathrm{b}\sqrt{3} \\ \Rightarrow 11-6\sqrt{3}=\mathrm{a}-\mathrm{b}\sqrt{3} \\ \Rightarrow \mathrm{a}=11;\mathrm{b}=6 \end{gathered}$$