GCSE Maths Practice: powers-and-roots

Understanding Fractional Indices

At Higher GCSE level, you are expected to interpret and simplify expressions with fractional powers confidently. Fractional indices combine two operations — taking a root and raising to a power. The rule that connects them is \(a^{m/n} = \sqrt[n]{a^m}\). This means you can either raise the number to the power m first and then take the nth root, or take the nth root first and then raise the result to the power m.

The Power of a Power Law

When an expression already involves an exponent, such as \((a^p)^q\), the law of indices says you multiply the exponents: \((a^p)^q = a^{p\times q}\). This rule works with all kinds of powers — integers, fractions, and negatives. It makes evaluating nested exponents much easier, since you can rewrite a complicated chain of powers as a single exponent.

Step-by-Step Method

1. Identify the inner and outer exponents in the expression.
2. Multiply them to find the overall fractional power.
3. Rewrite the fractional power as a root and a power using \(a^{m/n} = \sqrt[n]{a^m}\).
4. Simplify by calculating either the root or the power first — the order doesn’t matter for positive numbers.

Worked Examples (Different Numbers)

• \((3^2)^{1/2} = 3^{2/2} = 3^1 = 3\)
• \((5^3)^{1/2} = 5^{3/2} = \sqrt{125} \approx 11.18\)
• \((4^3)^{1/3} = 4^{3/3} = 4^1 = 4\)
• \((9^2)^{1/4} = 9^{2/4} = 9^{1/2} = 3\)

Each example uses the same principles: multiply the exponents and then simplify using root notation if needed.

Common Mistakes

• Reversing the exponent fraction — \(a^{1/3}\) is the cube root, not division by three.
• Forgetting to multiply exponents when applying the power-of-a-power rule.
• Using a calculator too early and rounding before simplification — always simplify symbolically first.
• Thinking that fractional exponents make numbers smaller in all cases — the effect depends on whether the base is greater or less than one.

Real-Life Applications

Fractional powers appear throughout science, finance, and computing. In physics, formulas for energy and light intensity use square or cube roots of quantities. In finance, compound interest over fractional years uses fractional exponents. In computer science, algorithms involving time complexity sometimes include powers like \(n^{1/2}\) or \(n^{2/3}\) to express growth rates. Learning to manipulate these exponents prepares you for advanced problem solving beyond GCSE.

Quick FAQ

• Q1: What does the numerator and denominator mean in a fractional exponent?
  A1: The numerator tells you the power; the denominator tells you which root to take.
• Q2: Does the order of taking the power and root matter?
  A2: For positive bases, no — both give the same result because of index laws.
• Q3: How do I recognise when to use this rule?
  A3: Any time a question mixes roots and powers, rewrite it as a fractional exponent before simplifying.

Study Tip

Practise converting between roots and fractional powers: \(\sqrt{a} = a^{1/2}\), \(\sqrt[3]{a} = a^{1/3}\), and \(a^{m/n} = (\sqrt[n]{a})^m\). When simplifying nested powers, multiply the exponents first, then evaluate or estimate. Always check your final answer by raising it back to the original power — if it reproduces the base expression, your calculation is correct.