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SQA National 5 Mathematics

Indices

Using index laws with positive, negative and fractional powers.

Formula referenceMixed revision

Formula reference

Check the National 5 rules and formulae linked to this topic.

Which method?

Match exam clues to a suitable method.

Before you start

  • Know the index laws for multiplying and dividing powers.
  • Remember that a⁰ = 1 when a ≠ 0.
  • Negative powers mean reciprocals.
National 5 Mathematics lesson

Explanation

Indices are powers. They show repeated multiplication, such as 2⁴ = 2 × 2 × 2 × 2

When the base is the same, use the index laws to simplify before calculating.

National 5 index questions often combine positive, zero, negative and fractional powers.

Key formulae and rules

  • am x an = am+n
  • am / an = am−n
  • (am)n = amn
  • a⁰ = 1
  • a−n = 1 / an

Watch out

Multiplying powers instead of adding them.

Check

Look for a square factor or matching base before reaching for a calculator.

Exam tip

Check whether the base is exactly the same before applying an index law.

Calculator tip

Keep exact form until the final answer. Use decimals only when the question asks for them.

Worked examples

Worked example 1

Multiply powers

Simplify x³ x x⁵.

  1. The base is the same.
  2. Add the powers: 3 + 5 = 8

Answer: x⁸

Worked example 2

Divide powers

Simplify a⁷ / a².

  1. The base is the same.
  2. Subtract the powers: 7 − 2 = 5

So: a⁵.

Worked example 3

Use a negative power

Simplify 3−2.

  1. A negative power means reciprocal.
  2. 3−2 = 1 / 3²
  3. 3² = 9

Answer: 19

Watch out

  • Multiplying powers instead of adding them.
  • Thinking a⁰ = 0
  • Forgetting brackets in expressions like (2x)³