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Algebra

Question 17 of 20

Surds

Question

Rationalise the denominator in \(\dfrac{2}{1-\sqrt{2}}\)

Show solution Hide solution Fully worked — 4 steps
  1. Multiply the numerator and denominator by \(1 + \sqrt{2}\)
    \[ \frac{2}{1-\sqrt{2}} = \frac{2}{1-\sqrt{2}} \times \frac{1+\sqrt{2}}{1+\sqrt{2}} \]
  2. Expand the brackets

    In the numerator \(2(1 + \sqrt{2}) = 2 + 2\sqrt{2}\), and in the denominator \((1 - \sqrt{2})(1 + \sqrt{2}) = 1 - 2\):

    \[ = \frac{2+2\sqrt{2}}{1-2} \]
  3. Evaluate the denominator

    Evaluating \(1 - 2\) as \(-1\):

    \[ = \frac{2+2\sqrt{2}}{-1} \]
  4. Final answer

    Dividing the numerator by \(-1\):

    \[ \frac{2}{1-\sqrt{2}} = -2 - 2\sqrt{2} \]

    That is, with the denominator rationalised, the answer is minus two, minus two root two.