GCSE Maths Practice: powers-and-roots

Question 6 of 10

This question checks your understanding of squaring negative numbers — a key part of the Powers and Roots topic in GCSE Maths.

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\( \begin{array}{l} \text{What is } (-5)^2? \end{array} \)

Choose one option:

Always use brackets when squaring negatives to include the sign correctly. Even powers make negatives positive.

Understanding Squares of Negative Numbers

When a negative number is squared, the result is always positive. This happens because multiplying two negative numbers produces a positive product. Understanding this rule is essential for GCSE questions involving powers and algebraic signs.

Concept Breakdown

Squaring a number means multiplying it by itself. The operation affects both the number and its sign. For example, if we square a positive number, the result is positive. If we square a negative number, the result is still positive because the two negative signs cancel each other.

Step-by-Step Method

  1. Write the number inside brackets to avoid mistakes with the sign.
  2. Multiply the number by itself, including the negative sign.
  3. Apply the rule that a negative multiplied by a negative equals a positive.

Using brackets is important — without them, the negative sign may be left out of the squaring process.

Worked Examples (Different Numbers)

  • \((-3)^2 = 9\)
  • \((-6)^2 = 36\)
  • \((-2)^2 = 4\)

In each example, the result is positive because the negative signs cancel out.

Common Mistakes

  • Writing \(-5^2\) instead of \((-5)^2\). The first expression means the square of 5 with a negative sign in front (–25), while the second means the number –5 squared (25).
  • Forgetting to include brackets in calculations involving negatives.
  • Assuming all powers of negatives remain negative — only odd powers keep the negative sign.

Real-Life Applications

Squared values appear frequently in physics, engineering, and everyday problem solving. For example, when calculating energy or force where direction doesn’t matter, negative and positive values produce the same squared magnitude. Similarly, when working out areas, the square of a measurement must be positive, even if the original direction was reversed.

Quick FAQ

  • Q1: Why is the square of a negative number positive?
    A1: Because multiplying two negatives results in a positive value.
  • Q2: What happens if we square a negative number with no brackets?
    A2: Without brackets, the negative sign is not squared, leading to a different result.
  • Q3: Do higher even powers behave the same?
    A3: Yes, any even power of a negative number gives a positive result.

Study Tip

Always use brackets when squaring negative numbers. Remember: even powers of negatives become positive, while odd powers stay negative. Practise by listing values of \((-2)^n\) for n = 1, 2, 3, 4 to see the pattern.