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Algebra

Question 12 of 20

Surds

Question

Write \(\sqrt{\dfrac{32}{144}}\) in the simplified form \(a\sqrt{n}\)

Show solution Hide solution Fully worked — 4 steps
  1. Split the fraction into a quotient of roots

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

    \[ \sqrt{\frac{32}{144}} = \frac{\sqrt{32}}{\sqrt{144}} \]
  2. Split \(\sqrt{32}\) into perfect-square factors

    Using the rule \(\sqrt{(a \times b)} = \sqrt{a} \times \sqrt{b}\), and evaluating \(\sqrt{144}\) as \(12\):

    \[ = \frac{\left(\sqrt{16} \times \sqrt{2}\right)}{12} \]
  3. Evaluate \(\sqrt{16}\)
    \[ = \frac{4\sqrt{2}}{12} \]
  4. Final answer

    Simplifying \(\frac{4}{12}\):

    \[ \sqrt{\frac{32}{144}} = \frac{\sqrt{2}}{3} \]

    That is, in the simplified form, the answer is the square root of two over three.