GCSE Maths Practice: powers-and-roots

Exact Form with Fractional Indices

Fractional indices combine roots and powers in one notation. The key identity is a^m/n = \(\sqrt[n]{a^m}\). For Higher GCSE, you should be able to convert, simplify, and present answers in exact form (using surds) without switching to decimals.

Two Useful Views of a^3/2

• Product form: \(a^{3/2} = a^{1+1/2} = a\cdot a^{1/2} = a\sqrt{a}\).
• Root–power form: \(a^{3/2} = (\sqrt{a})^3\).

Both approaches are equivalent for positive a. Choose the one that leads to quicker simplification.

How to Keep Results Exact

1. Rewrite the fractional power using either product or root–power form.
2. Simplify any square roots by factoring out perfect squares (e.g., \( \sqrt{36b}=6\sqrt{b} \)).
3. Rationalise only if a surd appears in a denominator (not needed if your answer is already in the numerator).
4. Avoid approximations unless the question explicitly asks for decimals.

Worked Patterns (Different Bases)

• \(k^{3/2} = k\sqrt{k}\) (product view) — fastest when \(\sqrt{k}\) simplifies partly.
• \(m^{3/2} = (\sqrt{m})^3\) (root–power view) — helpful when \(\sqrt{m}\) is easy to rewrite like \(c\sqrt{d}\).
• Examples (with different numbers): \(7^{3/2}=7\sqrt{7}\), \(18^{3/2}=(\sqrt{18})^3=(3\sqrt{2})^3=27\cdot 2\sqrt{2}=54\sqrt{2}\).

Common Mistakes

• Mixing up numerator and denominator in \(a^{m/n}\). Remember: denominator = root, numerator = power.
• Switching to decimals early and losing exactness.
• Forgetting to simplify the surd (e.g., leaving \(\sqrt{20}\) instead of \(2\sqrt{5}\)).
• Confusing \(a^{3/2}\) with \((a^3)^2\) — use index laws carefully.

Why Exact Form Matters

Exact surd answers preserve precision and are standard in GCSE algebra and surds work. They also link directly to later topics (e.g., solving quadratics that produce surds, working with compound indices, and simplifying expressions for calculus in further study).

Quick FAQ

• Do I take the root or power first? For positive bases it doesn’t matter; use the route that simplifies faster.
• When should I rationalise? Only when a surd is in the denominator of a fraction.
• How do I spot perfect-square factors? Split the radicand into prime factors or known squares (4, 9, 16, 25, …).

Study Tip

Practise rewriting \(a^{3/2}\) in both ways and always finish by simplifying the surd. Build a quick mental list of perfect squares so you can extract them instantly from inside square roots.