GCSE Maths Practice: powers-and-roots

Question 10 of 10

This question tests your understanding of squaring fractions — a key part of the Powers and Roots topic at GCSE Foundation level.

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\( \begin{array}{l} \text{What is } \left(\tfrac{1}{2}\right)^2? \end{array} \)

Choose one option:

When squaring a fraction, square both the top and bottom numbers. The result is always smaller than one if the fraction is less than one.

Understanding How to Square Fractions

Squaring a fraction means multiplying it by itself. The same rule used for whole numbers applies to fractions — you simply multiply both the numerators together and both the denominators together. This skill is important in GCSE Maths because fractions appear in many topics such as ratios, proportions, and algebraic manipulation.

Concept Overview

When you square a number, you raise it to the power of two. The operation shows how many equal parts make up the whole when something is enlarged or scaled. For fractions, squaring makes the result smaller, since you are multiplying a number less than one by itself.

Step-by-Step Method

  1. Write the fraction clearly inside brackets before applying the power.
  2. Multiply the top (numerator) by itself.
  3. Multiply the bottom (denominator) by itself.
  4. Simplify the result if necessary.

Worked Examples (Different Numbers)

  • \(\left(\tfrac{3}{4}\right)^2 = \tfrac{9}{16}\)
  • \(\left(\tfrac{2}{3}\right)^2 = \tfrac{4}{9}\)
  • \(\left(\tfrac{5}{6}\right)^2 = \tfrac{25}{36}\)

Notice that in each case, the fraction becomes smaller when squared. This is because multiplying two numbers less than one always gives a smaller product.

Common Mistakes

  • Squaring only the numerator or only the denominator instead of both.
  • Adding instead of multiplying the fractions.
  • Forgetting to include brackets, which can change the meaning of the expression.

Real-Life Applications

Squaring fractions is used when finding areas of shapes with fractional dimensions. For example, if the side of a square field is half a metre, its area is found by squaring that fraction. This also appears in scale models, recipes, and probability problems where parts of a whole are repeated or combined.

Quick FAQ

  • Q1: Does squaring a fraction make it bigger or smaller?
    A1: If the fraction is less than one, the square is smaller than the original fraction.
  • Q2: How do I check my result?
    A2: Multiply the fraction by itself and simplify.
  • Q3: Can mixed numbers be squared the same way?
    A3: Yes, but it’s easier to convert them to improper fractions first.

Study Tip

Always use brackets when squaring fractions or algebraic terms. Practise squaring common fractions like ½, ⅔, and ¾ to build confidence for GCSE calculations involving ratios, scaling, and area.