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1.6 Index Notation

GCSE Maths Foundation · Edexcel · Chapter 1: Number

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: writing and working out powers, and the index laws for multiplying and dividing powers. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Write repeated multiplication in index form and work out the value of a power
  • Use powers of 10 and the order of operations with powers
  • Multiply and divide powers of the same number by adding or subtracting indices
  • Use the rules for a power of a power and for a power of 0

Key words

  • Index form: A short way of writing repeated multiplication: 3 × 3 × 3 × 3 = 3⁴.
  • Base: The number being multiplied. In 3⁴ the base is 3.
  • Index (power): How many times the base is used. In 3⁴ the index is 4. The plural is indices.
  • Index laws: Rules for combining powers of the same base.

Writing in index form

Each time the index goes up by 1, you multiply by the base again. Powers of 2 double each time:

Worked example: index form

(a) 6 × 6 × 6 = 6³
(b) 2 × 2 × 2 × 2 × 2 = 2⁵ = 32
(c) 3 × 3 × 7 × 7 × 7 = 3² × 7³

Answer: 6³, 2⁵ = 32, 3² × 7³

Working out powers

Powers come before ×, ÷, + and − (BIDMAS). A negative number to an even power is positive; to an odd power it is negative.

  • Value; 10¹ 10; 10² 100; 10³ 1000; 10⁴ 10 000; 10⁵ 100 000; 10⁶ 1 000 000

Worked example: order of operations

(a) 3 × 2⁴ = 3 × 16 = 48
(b) (3 × 2)⁴ = 6⁴ = 1296
(c) (−2)³ = −2 × −2 × −2 = −8 (d) (−1)⁶ = 1

Answer: 48, 1296, −8, 1

Multiplying and dividing powers

Write out 7³ × 7² in full: (7 × 7 × 7) × (7 × 7) = 7⁵. You add the indices. Dividing cancels: 6⁵ ÷ 6² = 6³. You subtract the indices.

  • am × aⁿ = am⁺ⁿ; In words multiplying: add the indices; Example 7³ × 7⁵ = 7⁸
  • am ÷ aⁿ = am−ⁿ; In words dividing: subtract the indices; Example 6⁹ ÷ 6⁴ = 6⁵

Worked example: mixed

Write 4⁶ × 4³ ÷ 4⁵ as a single power of 4.
4⁶ × 4³ = 4⁹
4⁹ ÷ 4⁵ = 4⁴

Answer: 4⁴

Key fact: The laws only work when the bases are the same. 2³ × 5² cannot be written as one power: work it out, 8 × 25 = 200.

Power of a power, and the power 0

(3²)⁴ means 3² × 3² × 3² × 3² = 3⁸. For a power of a power, multiply the indices.

Any number divided by itself is 1. So 9³ ÷ 9³ = 1, but the index law gives 9³−³ = 9⁰. That is why any number (except 0) to the power 0 is 1.

Worked example: power of a power and zero index

(a) (5³)² = 5³×² = 5⁶
(b) 8⁰ + 3² = 1 + 9 = 10
(c) Write 9² as a power of 3: 9 = 3², so 9² = (3²)² = 3⁴

Answer: 5⁶, 10, 3⁴

Common mistakes

  • Multiplying the base by the index: 3⁴ is 81, not 12.
  • Multiplying the bases: 2³ × 2⁴ = 2⁷, not 4⁷.
  • Multiplying indices when you should add: 5³ × 5⁴ = 5⁷, not 5¹².
  • Thinking 7⁰ = 0 or 7⁰ = 7. Any number to the power 0 is 1.

Revision questions

Try these, then check your answers below each one.

Question 1. Work out 2⁵.

Answer: 32

Working: 2 × 2 × 2 × 2 × 2 = 32.

Question 2. Write 5⁴ × 5³ as a single power of 5. (Choose from: 5⁷, 5¹², 25⁷, 5¹.)

Answer: 5⁷

Working: 5⁴⁺³ = 5⁷.

Question 3. Write 8⁹ ÷ 8³ as a single power of 8. (Choose from: 8⁶, 8³, 8¹², 1⁶.)

Answer: 8⁶

Working: 8⁹⁻³ = 8⁶.

Related lessons

Continue in the full course

This lesson is section 1.6 (EGCMF0106) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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