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Algebra

Question 15 of 18

Indices & the Law of Indices

Question

Simplify \(\sqrt[5]{\dfrac{x^{10}}{32y^{25}}}\)

Show solution Hide solution Fully worked — 3 steps
  1. Apply the fractional-index law \(\sqrt[n]{a^{m}} = a^{m/n}\) together with the power-of-a-product law \((ab)^{n} = a^{n}b^{n}\) and the power-of-a-quotient law \(\left(\frac{a}{b}\right)^{n} = \frac{a^{n}}{b^{n}}\)
    \[ \sqrt[5]{\frac{x^{10}}{32y^{25}}} = \left(\sqrt[5]{\frac{1}{32}}\right)\left(\frac{x^{10/5}}{y^{25/5}}\right) \]
  2. Evaluate \(\sqrt[5]{\frac{1}{32}}\)

    The indices \(10/5\) and \(25/5\) evaluate to \(2\) and \(5\):

    \[ = \left(\frac{1}{2}\right)\left(\frac{x^{2}}{y^{5}}\right) \]
  3. Multiply the fractions
    \[ \sqrt[5]{\frac{x^{10}}{32y^{25}}} = \frac{x^{2}}{2y^{5}} \]

    That is, the simplified expression is x squared, all over 2 y to the power 5.