---
title: "Zero and Negative Indices"
description: "Extend the index laws so that any non-zero number to the power 0 is 1 and a negative index means the reciprocal."
canonical: https://lightmysky.com/learn/mathematics/zero-and-negative-indices-mt_jAtXBSVqkj
source: https://lightmysky.com/learn/mathematics/zero-and-negative-indices-mt_jAtXBSVqkj.md
retrieved: 2026-09-12
---

> **Agent view.** This is the Markdown twin of the page, for tools and assistants.
> When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md
> API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md
> Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt
> Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json
> Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`.

# Zero and Negative Indices

Extend the index laws so that any non-zero number to the power 0 is 1 and a negative index means the reciprocal.

Subject: Mathematics · Area: Number Representation & Place Value · Ages 14 to 15
Page: https://lightmysky.com/learn/mathematics/zero-and-negative-indices-mt_jAtXBSVqkj

## Ready when they can

- Rewrite an expression with negative indices in fraction form
- Explain why x⁰ = 1 by dividing two equal powers
- Combine the multiplication, division and power laws in one expression

## Lesson: Taming zero and negative powers

Take x to the 5th divided by x to the 5th. Any number divided by itself is 1. The quotient rule turns the same division into x to the power 5 minus 5, which is x to the 0. Both ways describe one division, so x to the 0 must equal 1, for any nonzero number.

Now try x cubed divided by x to the 5th. The quotient rule gives x to the power 3 minus 5, which is x to the minus 2. Cancelling by hand leaves 1 over x squared. So a negative index means flip into a fraction and drop the minus sign: x to the minus 3 is 1 over x cubed.

**Example.** Numbers work the same way. 5 to the minus 2 is 1 over 25, which is 0.04, and 2 to the minus 3 is 1 over 8, which is 0.125. With letters, multiply 2 x to the minus 4 by 3 x to the 6th: the numbers give 6 and the indices add to 2, so the answer is 6 x squared. When a power is raised to another power, you multiply the indices, so a to the minus 3 squared is a to the minus 6.

**Tip.** Keep the old laws running. Add indices when multiplying and subtract when dividing, then flip any negative leftover into a fraction. Try y to the minus 5 divided by y to the minus 2: subtract to get y to the minus 3, which is 1 over y cubed.

**Recap.** Zero power is 1, negative powers flip, and the old index laws keep working.

## Practice

18 questions on this page, each with its working shown.

## Needs first

- [Laws of Exponents](https://lightmysky.com/learn/mathematics/laws-of-exponents-mt_UjNba32kkz)

## Opens up

- [The Laws of Logarithms](https://lightmysky.com/learn/mathematics/the-laws-of-logarithms-mt_5vHdWlxaqY)
- [Fractional Indices and Roots](https://lightmysky.com/learn/mathematics/fractional-indices-and-roots-mt_fdK-nbQcBu)
- [Standard Form](https://lightmysky.com/learn/mathematics/standard-form-mt_GbZZ2xjBTt)
- [pH, Kw and Strong Acids and Bases](https://lightmysky.com/learn/science/ph-kw-and-strong-acids-and-bases-mt_KJ4yc25Z_l)