Rationalise the denominator of (2√2 − √3)/(√3 + √2).

Step-by-step solution

1. 1

   Identify the conjugate of the denominator. For a denominator $\sqrt a+\sqrt b$ the conjugate is $\sqrt a-\sqrt b$, because their product is the rational number $a-b$. Here the conjugate of $\sqrt3+\sqrt2$ is $\sqrt3-\sqrt2$.

2. 2

   Multiply numerator and denominator by that conjugate.

   $\frac{2\sqrt2-\sqrt3}{\sqrt3+\sqrt2}\cdot\frac{\sqrt3-\sqrt2}{\sqrt3-\sqrt2}$

3. 3

   Simplify the denominator first — it is the easy half.

   $\left(\sqrt3+\sqrt2\right)\left(\sqrt3-\sqrt2\right)=3-2=1$

   A denominator of $1$ means the whole answer will be just the expanded numerator.

4. 4

   Expand the numerator.

   $\left(2\sqrt2-\sqrt3\right)\left(\sqrt3-\sqrt2\right)=2\sqrt6-2\cdot 2-3+\sqrt6=2\sqrt6-4-3+\sqrt6$

   using $\sqrt2\cdot\sqrt3=\sqrt6$, $\sqrt2\cdot\sqrt2=2$ and $\sqrt3\cdot\sqrt3=3$.

5. 5

   Collect like terms.

   $3\sqrt6-7$

   $\boxed{3\sqrt6-7}$

6. 6

   Check numerically. $\sqrt2=1.41421$, $\sqrt3=1.73205$, so the original is $\dfrac{2.82843-1.73205}{1.73205+1.41421}=\dfrac{1.09637}{3.14626}=0.34846$, and $3\sqrt6-7=3(2.44949)-7=7.34847-7=0.34847$ ✓.