Expression with rationalisations

Statement

Compute $-20\sqrt{2}+\left(\frac{1+\sqrt{2}}{1-\sqrt{2}}\right)^2 - \frac{16}{2+\sqrt{2}}.$

Solution

We rationalise each piece:

\frac{1+\sqrt{2}}{1-\sqrt{2}} &= \frac{(1+\sqrt{2})^2}{(1-\sqrt{2})(1+\sqrt{2})} = \frac{3+2\sqrt{2}}{-1} = -3-2\sqrt{2} \\[4pt] \frac{16}{2+\sqrt{2}} &= \frac{16(2-\sqrt{2})}{4-2} = 16-8\sqrt{2} \end{aligned}$$ Then: $$-20\sqrt{2} + (-3-2\sqrt{2})^2 - (16-8\sqrt{2}) = -20\sqrt{2} + 9+12\sqrt{2}+8 - 16+8\sqrt{2} = \boxed{1}$$

Links

Topics: Radicals
Concepts: Conjugate · Rationalisation
Methods: Radical rationalisation
Skills: Calculating · Simplifying
Exercise type: Expression calculation