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Indices and standard form
Lesson 7 of 75
Easy practice

Zero and negative indices

Objective. Interpret zero and negative integer powers.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

Zero and negative indices extend the index laws beyond positive whole-number powers. Any non-zero number raised to the power zero equals 1, so a⁰ = 1, which follows from aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰ = 1. A negative index means the reciprocal of the positive power, so a⁻ⁿ = 1 ÷ aⁿ. For example, 5⁻² means 1 ÷ 5² = 1/25. These rules keep the index laws consistent when subtraction of indices produces zero or a negative result, such as a³ ÷ a⁵ = a⁻².

Worked example

Evaluate 4⁰ + 2⁻³ as a single fraction.

4 … 2
Illustrated card showing 4 … 2
  1. 14⁰ = 1, since any non-zero number to the power zero is 1.
  2. 22⁻³ = 1 ÷ 2³ = 1/8.
  3. 3Add: 1 + 1/8 = 8/8 + 1/8.
  4. 48/8 + 1/8 = 9/8.

Answer: 9/8

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Common mistake. Pupils often think a negative index makes the value itself negative, writing 2⁻³ as −8 instead of 1/8. A negative index means 'reciprocal', not 'negative number'.

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