Question

Find the values of a and b in each of the following.
$\frac{5-\sqrt{6}}{5+\sqrt{6}}=a-b\sqrt{6}$

Answer 1

We have:
$\frac{5-\sqrt{6}}{5+\sqrt{6}}$

Now, rationalising the denominator of the given number by multiplying both the numerator and denominator with $5-\sqrt{6}$, we get:
$$\begin{gathered} \frac{5-\sqrt{6}}{5+\sqrt{6}}=\frac{5-\sqrt{6}}{5+\sqrt{6}}\times \frac{5-\sqrt{6}}{5-\sqrt{6}} \\ =\frac{{\left(5-\sqrt{6}\right)}^2}{{\left(5\right)}^2-{\left(\sqrt{6}\right)}^2} \\ =\frac{25+6-10\sqrt{6}}{25-6} \\ =\frac{31-10\sqrt{6}}{19} \\ =\frac{31}{19}-\frac{10}{19}\sqrt{6} \end{gathered}$$

∴ $\frac{31}{19}-\frac{10}{19}\sqrt{6}$ = a $-$ b$\sqrt{6}$
So, on comparing the LHS and the RHS, we get:
a = $\frac{31}{19}$ and b = $\frac{10}{19}$