Fractional Indices

Statement

Fractional indices allow us to express roots using powers. The key rules are:

\[ a^{1/n} = \sqrt[n]{a}, \quad a^{m/n} = \sqrt[n]{a^m}, \quad (a > 0) \]

Why it works

• The index (exponent) of 1/2 means "square root", so \(a^{1/2} = \sqrt{a}\).
• The index of 1/3 means "cube root", so \(a^{1/3} = \sqrt[3]{a}\).
• More generally, an index of 1/n means the n-th root: \(a^{1/n} = \sqrt[n]{a}\).
• If the power is m/n, the denominator gives the root and the numerator gives the power: \[a^{m/n} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}\].

Recipe (Steps to Apply)

1. Look at the fraction in the power: numerator = power, denominator = root.
2. First apply the root (denominator), then the power (numerator).
3. Simplify step by step to avoid mistakes.

Spotting it

Fractional indices appear when converting between radical form and index form, and when solving exponential equations that involve roots.

Common pairings

• Simplifying surds and powers.
• Solving exponential equations.
• Using laws of indices (multiplication, division, powers).

Mini examples

1. \(9^{1/2} = \sqrt{9} = 3\).
2. \(8^{1/3} = \sqrt[3]{8} = 2\).
3. \(27^{2/3} = \left(\sqrt[3]{27}\right)^2 = 3^2 = 9\).

Pitfalls

• Mixing up numerator and denominator — remember denominator = root, numerator = power.
• Forgetting that roots must be positive real numbers in GCSE-level work (e.g. \(a > 0\)).
• Applying the power before the root incorrectly — always do root first, then power.

Exam Strategy

• Rewrite roots as fractional powers to apply laws of indices easily.
• Check that your simplified form makes sense (e.g. cube root of 8 = 2, not 4).
• If unsure, rewrite as a radical and calculate step by step.

Worked Micro Example

Evaluate \(16^{3/4}\).

Step 1: Denominator = 4 → fourth root: \(\sqrt[4]{16} = 2\).

Step 2: Raise to numerator 3: \(2^3 = 8\).

So \(16^{3/4} = 8\).

Summary

Fractional indices express roots as powers, making it easier to apply the laws of indices. The denominator of the fraction is the root, the numerator is the power. This is essential in algebra, simplifying expressions, and solving equations.