GCSE Maths Practice: powers-and-roots

Understanding Fourth Roots and Fractional Indices

At higher levels of GCSE Maths, you are expected to understand that powers and roots are part of the same system. A root can always be expressed as a fractional power. The fourth root of a number means the value that, when multiplied by itself four times, equals the original number. Using indices, this is written as \(a^{1/4}\).

How Roots and Powers Are Connected

In the language of indices, \(a^2\) represents squaring, \(a^3\) represents cubing, and \(a^{1/2}\) represents taking the square root. Extending this pattern, \(a^{1/3}\) gives the cube root, and \(a^{1/4}\) gives the fourth root. This unified rule helps simplify even complex expressions in algebra and allows you to apply the same laws of indices to all powers, including fractional and negative ones.

Step-by-Step Method

1. Rewrite any root as a fractional index. For example, the fourth root of a number becomes \(a^{1/4}\).
2. Identify whether the base can be expressed as a smaller number to a known power, such as \(2^4\), \(3^2\), or \(5^3\).
3. Apply index laws: when raising a power to another power, multiply the exponents.
4. Simplify by evaluating or comparing the powers to confirm the relationship between the base and its root.

Worked Examples (Different Numbers)

• \(\sqrt[4]{16} = 2\) because \(2^4 = 16\).
• \(\sqrt[4]{81} = 3\) because \(3^4 = 81\).
• \(\sqrt[4]{625} = 5\) because \(5^4 = 625\).
• \(\sqrt[4]{10,000} = 10\) because \(10^4 = 10,000\).

All these examples follow the same rule: the nth root reverses raising a number to the nth power.

Common Mistakes

• Confusing \(\sqrt[4]{a}\) with \(\sqrt{a}\). A fourth root is not the same as a square root — it is a deeper root that produces a smaller result.
• Forgetting that fractional indices represent roots, not divisions.
• Applying square-root logic incorrectly to higher roots or misusing the laws of indices.
• Neglecting to consider negative or zero bases when applicable. Even roots of negative numbers are not real for GCSE-level work.

Real-Life Applications

Fourth roots appear in advanced physics, statistics, and engineering. In electrical power formulas, for instance, intensity relationships can depend on higher-order roots. In statistics, variance and standard deviation involve square roots, while more complex error models use fourth roots to normalise data. Understanding how to manipulate fractional indices helps in later study of algebraic functions and exponential equations.

Quick FAQ

• Q1: What does the number in the root symbol mean?
  A1: It tells you how many identical factors multiply to make the original number. For example, the cube root involves three factors, the fourth root involves four.
• Q2: How do I express a fourth root using indices?
  A2: Use the fractional index \(a^{1/4}\). This allows you to apply the same rules as other powers.
• Q3: Can I apply index laws with fractional powers?
  A3: Yes — the laws work the same way. For example, \((a^{1/4})^2 = a^{1/2}\).

Study Tip

Memorise the relationship between powers and roots: \(a^{1/2}\) = square root, \(a^{1/3}\) = cube root, \(a^{1/4}\) = fourth root, and \(a^{m/n} = \sqrt[n]{a^m}\). When simplifying expressions, always convert roots into fractional powers before using the laws of indices. Practising this method prepares you for more complex surd and exponential equations in A-level maths and beyond.