Question

find the value of' a' and ' b' of following 5 + 2 * route 3 / 7 + 4 * route 3 = a+b * route 3

Answer 1

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