Rationalise a denominator using the conjugate

How to use it

1. 1The conjugate of $a+\sqrt{b}$ is $a-\sqrt{b}$; of $\sqrt{a}+\sqrt{b}$ it is $\sqrt{a}-\sqrt{b}$.
2. 2The denominator becomes a difference of squares, which is rational.
3. 3Telescoping sums of $\frac{1}{\sqrt{n}+\sqrt{n+1}}$ collapse after rationalising each term.

Step-by-step

1. 1
   $\frac{1}{3+\sqrt{2}}\times \frac{3-\sqrt{2}}{3-\sqrt{2}}$

   Why: Multiply by the conjugate over itself, which is $1$.

2. 2
   $=\frac{3-\sqrt{2}}{9-2}=\frac{3-\sqrt{2}}{7}$