GCSE Maths Practice: powers-and-roots

Understanding Fractional Indices

Fractional indices are a natural extension of integer powers and roots. Instead of treating powers and roots as separate ideas, they can be expressed using one general rule: \(a^{m/n} = \sqrt[n]{a^m}\). This means that a fractional index combines two operations — the numerator represents a power, and the denominator represents the root. This notation allows us to handle complex expressions involving roots and powers using a single consistent rule.

The Connection Between Roots and Powers

Consider how squaring and square roots relate. If \(a^2\) squares a number, then \(a^{1/2}\) must reverse that operation, giving the square root. Likewise, \(a^{1/3}\) represents the cube root. By extension, \(a^{2/3}\) means cube root first, then square, or equivalently \((\sqrt[3]{a})^2\). The order of these steps does not matter for positive numbers, because multiplication of exponents is commutative.

Step-by-Step Method for Simplifying

1. Rewrite any root expression using fractional index form: replace \(\sqrt[n]{a^m}\) with \(a^{m/n}\).
2. Express the base as a power of a smaller prime if possible — this helps when simplifying.
3. Apply the law of indices: when raising a power to another power, multiply the exponents.
4. Finally, evaluate the resulting power, simplifying if needed.

Worked Examples (Different Numbers)

• \(\sqrt[3]{27^2} = 27^{2/3} = (3^3)^{2/3} = 3^{2} = 9\)
• \(\sqrt[4]{16^3} = 16^{3/4} = (2^4)^{3/4} = 2^3 = 8\)
• \(\sqrt[5]{32^2} = 32^{2/5} = (2^5)^{2/5} = 2^2 = 4\)

Each example uses the same pattern: convert to fractional form, simplify the base if possible, multiply exponents, and then evaluate.

Common Mistakes

• Reversing numerator and denominator in the exponent (\(a^{m/n}\) ≠ \(a^{n/m}\)).
• Forgetting that the denominator represents the root — not division by n.
• Ignoring the need to multiply exponents when a base itself is already a power.
• Using a calculator prematurely, missing chances for exact simplification.

Real-Life Applications

Fractional indices are not just symbolic. They appear in many real-world contexts: exponential growth, radioactive decay, and geometric scaling all use fractional powers. For instance, in physics, if the intensity of light varies with distance according to \(I = k d^{-2}\), the negative fractional power represents an inverse-square law. Similarly, in finance, compound growth over fractional time periods is expressed using fractional exponents.

Quick FAQ

• Q1: What does the numerator and denominator mean in \(a^{m/n}\)?
  A1: The numerator m tells you the power, and the denominator n tells you which root to take.
• Q2: Do I take the root first or the power first?
  A2: For positive numbers, it does not matter. Mathematically, both orders give the same result.
• Q3: Why use fractional indices at all?
  A3: They make algebraic manipulation easier — you can use index laws for both powers and roots in one form.

Study Tip

Memorise the basic conversions: \(a^{1/2} = \sqrt{a}\), \(a^{1/3} = \sqrt[3]{a}\), \(a^{2/3} = (\sqrt[3]{a})^2\). When facing large roots or powers, always rewrite the expression in fractional form before simplifying. This approach works for surds, cube roots, and even negative powers of roots. Mastery of fractional indices forms the foundation for higher topics such as exponential equations and logarithms.