Zero and Negative Indices: KS3 Maths

Why does any number to the power 0 equal 1?

The index division law states: aᵐ ÷ aⁿ = aᵐ⁻ⁿ.

Apply this when m = n: aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰.

But any number divided by itself equals 1: aⁿ ÷ aⁿ = 1.

Therefore a⁰ = 1 for any non-zero value of a.

Examples:

• 7⁰ = 1
• (−3)⁰ = 1
• (2/5)⁰ = 1
• 1 000 000⁰ = 1

The only exception is 0⁰, which is undefined (and not tested at KS3).

What does a negative index mean?

A negative index means take the reciprocal and change the sign of the power:

a⁻ⁿ = 1/aⁿ

This also follows from the index division law. Consider a² ÷ a⁵:

a² ÷ a⁵ = a²⁻⁵ = a⁻³.

But writing this out in full: a² ÷ a⁵ = (a × a) ÷ (a × a × a × a × a) = 1/a³.

So a⁻³ = 1/a³. ✓

How do you evaluate expressions with negative indices?

1. Flip the base to its reciprocal (write 1 over the base).
2. Apply the positive version of the power.

Worked examples:

Notice the last row: a fraction raised to a negative power flips the fraction and applies the positive power. (1/3)⁻² = (3/1)² = 9.

How do you work with negative indices in algebra?

The same rule applies to algebraic bases:

x⁻¹ = 1/x and x⁻² = 1/x²

Example: Write 3x⁻² as a fraction.

3x⁻² = 3 × (1/x²) = 3/x²

The coefficient (3) stays where it is — only the x⁻² part becomes a fraction.

Example: Write 1/(5a³) using a negative index.

1/(5a³) = (1/5) × a⁻³ = 0.2a⁻³ or equivalently (1/5)a⁻³.

How do zero and negative indices combine with other index laws?

The index laws for multiplication and division still apply:

• a^m × a^n = a^(m+n) — works for any integers m and n, including 0 and negatives.
• a^m ÷ a^n = a^(m−n)

Worked example: Simplify x³ × x⁻⁵.

x³ × x⁻⁵ = x³⁺(⁻⁵) = x³⁻⁵ = x⁻² = 1/x²

Worked example: Simplify (2a⁴) ÷ (8a⁷).

(2a⁴) ÷ (8a⁷) = (2/8) × a⁴⁻⁷ = (1/4) × a⁻³ = a⁻³/4 or equivalently 1/(4a³).

What common mistakes should you avoid?

Mistake 1 — Thinking a⁻ⁿ means a negative number. A negative index gives a positive fraction (when a is positive). 2⁻³ = 1/8, not −8.

Mistake 2 — Applying the negative to the coefficient. In 3x⁻², only x is raised to the power −2. The 3 stays: the answer is 3/x², not 1/(3x²).

Mistake 3 — Writing 0⁰ = 1. This case is undefined; only non-zero bases give a⁰ = 1.

Mistake 4 — Forgetting that a⁻¹ = 1/a. This is the reciprocal. For example, 4⁻¹ = 1/4, not −4.

Frequently asked questions

Why does (1/2)⁻³ equal 8, not 1/8?

Because a negative index flips the base to its reciprocal before applying the power. The reciprocal of 1/2 is 2. So (1/2)⁻³ = 2³ = 8. This is consistent with the rule a⁻ⁿ = 1/aⁿ: (1/2)⁻³ = 1/(1/2)³ = 1/(1/8) = 8.

Does a⁰ = 1 work if a is a fraction?

Yes. (3/4)⁰ = 1, (−5/7)⁰ = 1, and so on, for any non-zero value. The zero index always returns 1, regardless of what the base is, as long as it is not zero.

How are negative indices used in standard form?

Very large or small numbers written in standard form can use negative powers of 10. For example, 0.0003 = 3 × 10⁻⁴. Here 10⁻⁴ = 1/10 000 = 0.0001. Understanding negative indices is therefore essential for working with very small quantities in standard form.

What is the difference between a negative index and a negative number?

A negative number is less than zero, such as −5 or −12. A negative index is a power, such as a⁻² = 1/a². The two things are unrelated: a⁻² is always positive (when a is positive), even though the index −2 looks negative. The minus sign is telling you to take a reciprocal, not to make the answer negative.

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