GCSE Maths Practice: powers-and-roots

Question 1 of 10

This problem tests your understanding of cube roots — the inverse operation of cubing. Cube roots frequently appear in GCSE questions involving powers, indices, and volume of cubes.

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\( \begin{array}{l} \text{What is the cube root of } 27? \end{array} \)

Choose one option:

Review perfect cubes up to 10³ to quickly recognise cube root patterns. Practise with both positive and negative numbers to build full confidence for exam questions.

Understanding Cube Roots

The cube root of a number is the inverse of cubing that number. When you cube a number, you multiply it by itself three times. For instance, 3 × 3 × 3 = 27. The cube root operation works in reverse — you start with the result (27) and determine which number was multiplied by itself three times to produce it. Thus, the cube root of 27 is 3.

Concept and Definition

The cube root of a number x is written as \(\sqrt[3]{x}\). It represents the value that, when cubed, equals x. Unlike square roots, cube roots can yield both positive and negative results. For example, \(\sqrt[3]{-8} = -2\), because \(-2 \times -2 \times -2 = -8\). This makes cube roots particularly useful when dealing with negative numbers or when solving equations involving odd powers.

Step-by-Step Method

  1. Identify the number you need to find the cube root of.
  2. List small integers and cube them until you match the target number.
  3. Check by multiplying the chosen number three times.
  4. If no exact cube root exists, use estimation or a calculator to approximate.

Example: To find \(\sqrt[3]{27}\): test 1³ = 1, 2³ = 8, 3³ = 27. Therefore, the answer is 3.

Worked Examples

  • Example 1: \(\sqrt[3]{8} = 2\) because \(2^3 = 8\).
  • Example 2: \(\sqrt[3]{64} = 4\) because \(4^3 = 64\).
  • Example 3: \(\sqrt[3]{-125} = -5\) because \((-5)^3 = -125\).

Common Mistakes

  • Confusing cube roots with square roots. Remember that cube roots use three factors, not two.
  • Forgetting that negative numbers have negative cube roots, since cubing a negative remains negative.
  • Using calculators incorrectly and rounding too early in non-perfect cubes.

Real-Life Applications

Cube roots are used to calculate the side length of cubes when the volume is known. For example, if a cube’s volume is 27 cm³, each edge measures \(\sqrt[3]{27} = 3\) cm. In physics, cube roots appear in density problems or when working with cubic relationships such as pressure, temperature, or scaling factors in 3D models.

Quick FAQ

  • Q1: Can cube roots be negative?
    A1: Yes. Because an odd number of negative factors results in a negative product.
  • Q2: What if the number isn’t a perfect cube?
    A2: Use estimation or a calculator. For instance, \(\sqrt[3]{20}\) ≈ 2.71.
  • Q3: Are cube roots always rational?
    A3: No, many cube roots (like \(\sqrt[3]{2}\)) are irrational numbers.

Study Tip

Memorize the first ten perfect cubes (1³ to 10³) to save time in exams. Recognizing values like 27, 64, 125, and 216 instantly improves problem-solving speed in topics such as indices, surds, and volume calculations.

Cube roots form the foundation for higher GCSE Maths concepts like fractional indices and solving polynomial equations. Mastering them ensures accuracy in algebraic manipulation and deeper mathematical understanding.