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Algebra

Question 5 of 20

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Question

Simplify \(\left(\sqrt{20} - \sqrt{5}\right)\left(\sqrt{20} + \sqrt{5}\right)\)

Show solution Hide solution Fully worked — 3 steps
  1. Recognise a difference of two squares

    Since \((x + y)(x - y) = x^{2} - y^{2}\), where \(x = \sqrt{20}\) and \(y = \sqrt{5}\):

    \[ \left(\sqrt{20} - \sqrt{5}\right)\left(\sqrt{20} + \sqrt{5}\right) = \left(\sqrt{20}\right)^{2} - \left(\sqrt{5}\right)^{2} \]
  2. Evaluate \(\left(\sqrt{20}\right)^{2}\) and \(\left(\sqrt{5}\right)^{2}\)
    \[ = 20 - 5 \]
  3. Final answer

    Subtracting 5 from 20:

    \[ \left(\sqrt{20} - \sqrt{5}\right)\left(\sqrt{20} + \sqrt{5}\right) = 15 \]

    That is, the simplified expression is fifteen.