GCSE Maths Practice: powers-and-roots

Question 2 of 10

This problem checks your understanding of cube roots — the reverse of cubing. Cube roots are part of the Powers and Roots topic in GCSE Maths and often appear in volume questions and problems involving indices.

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\( \begin{array}{l} \text{What is the cube root of } 64? \end{array} \)

Choose one option:

When finding cube roots, test small integers in order. 4 × 4 × 4 = 64, so 4 is the cube root. Practise spotting perfect cubes for faster recall during exams.

Understanding Cube Roots

The cube root is the inverse of cubing a number. When you cube a number, you multiply it by itself three times. For example, 4 × 4 × 4 = 64. The cube root operation works in reverse — you begin with the result (64) and ask: what number multiplied by itself three times gives this value? The answer is 4. This relationship is written as \(\sqrt[3]{64} = 4\).

Concept and Definition

In mathematics, the cube root of a number x is the value that, when raised to the power of three, equals x. The notation \(\sqrt[3]{x}\) or x1/3 represents the same concept. Cube roots are unique because they can produce both positive and negative results. For instance, \(\sqrt[3]{-8} = -2\) because multiplying three negative twos gives –8. This property makes cube roots essential in understanding powers and indices in GCSE Maths.

Step-by-Step Method

  1. Identify the number whose cube root you need to find.
  2. Start testing small integers to see which one cubed gives the original number.
  3. If it’s not a perfect cube, use estimation or a calculator.

In this case, we need the cube root of 64. Test small numbers: 3³ = 27 (too small), 4³ = 64 (perfect match). Therefore, \(\sqrt[3]{64} = 4\).

Worked Examples

  • Example 1: \(\sqrt[3]{8} = 2\) because \(2^3 = 8\).
  • Example 2: \(\sqrt[3]{27} = 3\) because \(3^3 = 27\).
  • Example 3: \(\sqrt[3]{125} = 5\) because \(5^3 = 125\).

Common Mistakes

  • Mixing up square roots and cube roots. Cube roots involve three multiplications, not two.
  • Forgetting that negative numbers have negative cube roots.
  • Thinking that every number has a simple cube root — some cube roots are decimals or irrational numbers.

Real-Life Applications

Cube roots appear in everyday contexts, especially when dealing with three-dimensional shapes. For instance, if the volume of a cube is 64 cm³, each side measures \(\sqrt[3]{64} = 4\) cm. Engineers and scientists use cube roots to determine side lengths, densities, or to scale up and down 3D models in computer graphics. They also appear in physics when solving equations involving volume or spatial relationships.

Quick FAQ

  • Q1: Can the cube root of a negative number be negative?
    A1: Yes. For example, \(\sqrt[3]{-27} = -3\).
  • Q2: Is \(\sqrt[3]{64}\) the same as 641/3?
    A2: Yes — fractional indices represent the same operation.
  • Q3: How do I know if a number is a perfect cube?
    A3: If its cube root is a whole number (like 1, 8, 27, 64, 125, ...), then it’s a perfect cube.

Study Tip

Memorize the first ten perfect cubes (1³ to 10³). They are 1, 8, 27, 64, 125, 216, 343, 512, 729, and 1000. This knowledge saves time in exams and helps in other topics like simplifying indices, solving equations, and estimating volumes. Remember that understanding cube roots is a foundation for later GCSE topics like fractional indices and surds. Once you can recognise perfect cubes quickly, you’ll solve these questions in seconds!