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Algebra

Question 13 of 20

Surds

Question

Write \(\sqrt{\dfrac{52}{169}}\) in the simplified form \(a\sqrt{n}\)

Show solution Hide solution Fully worked — 3 steps
  1. Split the fraction into a quotient of roots

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

    \[ \sqrt{\frac{52}{169}} = \frac{\sqrt{52}}{\sqrt{169}} \]
  2. Split \(\sqrt{52}\) into perfect-square factors

    Using the rule \(\sqrt{(a \times b)} = \sqrt{a} \times \sqrt{b}\), and evaluating \(\sqrt{169}\) as \(13\):

    \[ = \frac{\left(\sqrt{4} \times \sqrt{13}\right)}{13} \]
  3. Final answer

    Evaluating \(\sqrt{4}\) as \(2\):

    \[ \sqrt{\frac{52}{169}} = \frac{2\sqrt{13}}{13} \]

    That is, in the simplified form \(a\sqrt{n}\), \(a = \frac{2}{13}\) and \(n = 13\): the answer is two thirteenths times the square root of thirteen.