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Algebra

Question 9 of 20

Surds

Question

Express \((5\sqrt{2} - \sqrt{5})(\sqrt{2} + \sqrt{5})\) in the form \(a + b\sqrt{c}\)

Show solution Hide solution Fully worked — 5 steps
  1. Expand the brackets
    \[ (5\sqrt{2} - \sqrt{5})(\sqrt{2} + \sqrt{5}) = 5(\sqrt{2})^{2} + 5(\sqrt{2})(\sqrt{5}) - (\sqrt{5})(\sqrt{2}) - (\sqrt{5})^{2} \]
  2. Multiply the surds

    Using the rule \(\sqrt{a} \times \sqrt{b} = \sqrt{(a \times b)}\):

    \[ = 10 + 5\sqrt{2 \times 5} - \sqrt{5 \times 2} - 5 \]
  3. Simplify

    Subtract \(5\) from \(10\), and write \(\sqrt{2 \times 5}\) and \(\sqrt{5 \times 2}\) as \(\sqrt{10}\):

    \[ = 5 + 5\sqrt{10} - \sqrt{10} \]
  4. Collect the like surds

    Using the rule \(a\sqrt{c} \pm b\sqrt{c} = (a \pm b)\sqrt{c}\):

    \[ = 5 + (5 - 1)\sqrt{10} \]
  5. Final answer
    \[ (5\sqrt{2} - \sqrt{5})(\sqrt{2} + \sqrt{5}) = 5 + 4\sqrt{10} \]

    That is, in the required form \(a + b\sqrt{c}\), the answer is five plus four times the square root of ten.