GCSE Maths Practice: powers-and-roots

Question 5 of 10

This question checks your understanding of cubing negative numbers — an important part of the Powers and Roots topic in GCSE Maths.

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\( \begin{array}{l} \text{What is } (-2)^3? \end{array} \)

Choose one option:

Remember: an odd number of negatives gives a negative result. Always use brackets when calculating powers of negative numbers.

Understanding Cubes of Negative Numbers

Cubing a number means multiplying it by itself three times. When the base is negative, each multiplication affects the sign of the result. Two negative factors produce a positive value, but multiplying by a third negative flips the sign back to negative. This pattern shows why cubes of negative numbers remain negative.

Key Concept

Even powers remove negative signs because an even number of negatives cancel each other. Odd powers keep one negative factor unpaired, so the result stays negative. This distinction is crucial when simplifying powers or evaluating expressions in GCSE Maths.

Step-by-Step Method

  1. Write the base number with brackets to keep the sign attached.
  2. Multiply the number by itself twice more.
  3. Apply sign rules carefully: negative × negative = positive, and positive × negative = negative.

Worked Examples (Different Numbers)

  • \((-3)^3 = -27\)
  • \((-1)^3 = -1\)
  • \((-4)^3 = -64\)

All of these examples have negative results because cubing an odd number of negatives leaves one negative sign in the product.

Common Mistakes

  • Forgetting to include brackets, which can change the meaning of the expression.
  • Assuming the result must be positive because of squaring rules — cubes behave differently.
  • Incorrectly multiplying the signs in the wrong order.

Real-Life Applications

Cubed values appear when calculating volumes. For example, the volume of a cube is side³. In real situations, negative numbers can represent direction, change, or balance in contexts like physics or finance. Cubing a negative number might describe a reversal of direction in a three-dimensional context, where the magnitude remains but orientation changes.

Quick FAQ

  • Q1: Why do cubes of negatives stay negative?
    A1: Because multiplying three negative numbers leaves one negative sign unpaired.
  • Q2: How do even and odd powers differ?
    A2: Even powers give positive results; odd powers preserve the original sign.
  • Q3: What happens if brackets are left out?
    A3: Without brackets, only the number is cubed, not the sign, producing a wrong answer.

Study Tip

Practise squaring and cubing both positive and negative numbers. Notice that the sign only changes for odd powers. This habit prevents mistakes later in algebra and indices questions.