GCSE Maths Practice: powers-and-roots

Question 4 of 10

This problem checks your understanding of square roots of fractions — a key skill in GCSE Powers and Roots topics.

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\( \begin{array}{l} \text{What is the square root of } \tfrac{9}{16}? \end{array} \)

Choose one option:

Remember: the square root of a fraction equals the fraction of the square roots. Always simplify the result if possible.

Understanding Square Roots of Fractions

The square root of a fraction means finding a number that, when multiplied by itself, gives that fraction. A key rule is that the square root of a fraction equals the fraction of the square roots: \(\sqrt{\tfrac{a}{b}} = \tfrac{\sqrt{a}}{\sqrt{b}}\), provided both a and b are positive.

Step-by-Step Method

  1. Take the square root of the numerator.
  2. Take the square root of the denominator.
  3. Simplify if possible.

Example: \(\sqrt{\tfrac{9}{16}} = \tfrac{\sqrt{9}}{\sqrt{16}} = \tfrac{3}{4}.\)

Worked Examples

  • \(\sqrt{\tfrac{4}{9}} = \tfrac{2}{3}\)
  • \(\sqrt{\tfrac{1}{25}} = \tfrac{1}{5}\)
  • \(\sqrt{\tfrac{49}{100}} = \tfrac{7}{10}\)

Common Mistakes

  • Forgetting to take the root of both numerator and denominator.
  • Mixing up with dividing by 2 instead of square rooting.
  • Assuming negatives appear in roots of positive fractions.

Real-Life Connections

Square roots of fractions appear in measurement problems, such as converting scales or finding sides of similar shapes. For example, if an area is \(\tfrac{9}{16}\) m², each side is \(\tfrac{3}{4}\) m long.

FAQ

  • Q: Can I simplify before taking the root?
    A: Yes, reducing the fraction first makes calculations easier.
  • Q: Is \(\sqrt{\tfrac{9}{16}}\) the same as \((\tfrac{3}{4})^2\)?
    A: Yes — they are inverse operations.

Study Tip

Practise with fractions that have perfect square numerators and denominators. This helps you recognise patterns quickly in GCSE questions involving ratios, area, and Pythagoras.