How to Use Zero, Negative and Fractional Indices
This page explains how to use zero, negative, and fractional indices in GCSE and IGCSE Mathematics. You will learn the key index rules, how to simplify expressions using powers, and how indices link to roots and reciprocals. Worked examples are written in clear exam style to help you gain full method marks, followed by practice questions for revision.
What are Indices?
An index (plural indices) tells you how many times a base is multiplied by itself. For example, \(2^4 = 2 \times 2 \times 2 \times 2 = 16\). Indices compress repeated multiplication, simplify algebraic expressions, and connect powers with roots and reciprocals.
The Power Laws
These three laws underpin everything on this page. They work for any indices, including zero, negative, and fractional values.
\(a^m \times a^n = a^{m+n}\)
\(a^m \div a^n = a^{m-n}\)
\((a^m)^n = a^{mn}\)
The Three Special Cases
Zero and negative powers: look at what happens when you divide by the base each time. The index drops by 1 at every step:
Dividing by the base at each step drops the index by 1. At \(2^0\) you reach 1; below it the powers become reciprocals.
The pattern shows that \(a^0 = 1\) for any non-zero base, and that negative indices give reciprocals. Algebraically: \(a^n \div a^n = a^{n-n} = a^0 = 1\). Note that \(0^0\) is undefined at this level.
Fractional powers: link powers to roots. The denominator tells you which root to take; the numerator tells you which power to raise to. For example, \(16^{\tfrac{3}{4}} = \bigl(\sqrt[4]{16}\bigr)^3 = 2^3 = 8\).
\(a^0 = 1 \quad (a \neq 0)\)
\(a^{-n} = \dfrac{1}{a^n}\)
\(a^{\tfrac{1}{n}} = \sqrt[n]{a} \qquad a^{\tfrac{m}{n}} = \bigl(\sqrt[n]{a}\bigr)^m\)
Core Ideas
Worked Examples
💡 Example 1: Negative power
Evaluate \(2^{-3}\).
A negative exponent creates a reciprocal, not a negative value.
💡 Example 2: Fractional power
Evaluate \(16^{\tfrac{3}{4}}\).
Root first (denominator), then power (numerator).
💡 Example 3: Negative and fractional combined
Evaluate \(64^{-\tfrac{2}{3}}\).
What's happening?
The negative sign means "take the reciprocal".
Deal with the fraction: the denominator 3 means cube root, the numerator 2 means square.
Find the cube root first: \(\sqrt[3]{64} = 4\).
Then square: \(4^2 = 16\).
Finally apply the reciprocal: \(\dfrac{1}{16}\).
💡 Example 4: Fractional base
Evaluate \(\bigl(\tfrac{9}{16}\bigr)^{-\tfrac{1}{2}}\).
The negative index flips the fraction first; then the \(\tfrac{1}{2}\) means square root.
💡 Example 5: Simplify with power laws
Simplify \(\dfrac{3^5 \times 3^{-2}}{3^4}\).
Add indices when multiplying (same base), subtract when dividing.
🔑 Key Points
- \(x^{-1}\) means \(\tfrac{1}{x}\), not \(-x\).
- For \(a^{m/n}\): root first (denominator), then power (numerator).
- The standard power laws still apply: same base means add indices when multiplying, subtract when dividing.
⚠️ Common Pitfalls
- Confusing \(x^{-2}\) with \(-x^2\). The negative is in the exponent, not the base.
- Forgetting to root before powering with fractional indices. Always do the denominator first.
- Writing \(0^0 = 1\). At IGCSE level, \(0^0\) is undefined.
- Applying a negative index to the wrong part of a fraction, e.g. \(\bigl(\tfrac{2}{3}\bigr)^{-1} = \tfrac{3}{2}\), not \(\tfrac{-2}{3}\).