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Algebra

Question 6 of 20

Surds

Question

Simplify \(\dfrac{2\sqrt{3}}{5} + \sqrt{108}\)

Show solution Hide solution Fully worked — 5 steps
  1. Write both terms over a common denominator

    Find the lowest common denominator to add — here it is 5:

    \[ \frac{2\sqrt{3}}{5} + \sqrt{108} = \frac{2\sqrt{3} + 5\sqrt{108}}{5} \]
  2. Split \(\sqrt{108}\) into perfect-square factors

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

    \[ = \frac{2\sqrt{3} + 5\left(\sqrt{36} \times \sqrt{3}\right)}{5} \]
  3. Evaluate \(\sqrt{36}\) and multiply by 5
    \[ = \frac{2\sqrt{3} + 30\sqrt{3}}{5} \]
  4. Collect the like surds

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

    \[ = \frac{(2 + 30)\sqrt{3}}{5} \]
  5. Final answer
    \[ \frac{2\sqrt{3}}{5} + \sqrt{108} = \frac{32\sqrt{3}}{5} \]

    That is, the simplified expression is thirty-two times the square root of three, all over five.