18.2 The Laws of Indices

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: multiplying and dividing powers, a power of a power, the zero index and negative indices. 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

• Use index notation and work out powers of numbers
• Use the laws of indices to multiply and divide powers of the same number or letter
• Raise a power to a power, and use the zero index
• Work out negative powers as fractions (reciprocals)

Key words

• Index (power): The small number that shows how many times to multiply the base by itself. In 5³ the index is 3. Plural: indices.
• Base: The number or letter being multiplied. In 5³ the base is 5.
• Index form: A number written as a power, such as 2⁶ instead of 64.
• Reciprocal: 1 divided by the number. The reciprocal of 7 is 1/7.

Index notation

A power is short for repeated multiplication: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. Learn the common ones so you can spot them on a non-calculator paper.

• 2; power 1 2; power 2 4; power 3 8; power 4 16; power 5 32
• 3; power 1 3; power 2 9; power 3 27; power 4 81; power 5 243
• 5; power 1 5; power 2 25; power 3 125; power 4 625; power 5 3125
• 10; power 1 10; power 2 100; power 3 1000; power 4 10 000; power 5 100 000

Worked example: evaluating powers

(a) 4³ = 4 × 4 × 4 = 64 (b) (−2)⁴ = (−2) × (−2) × (−2) × (−2) = 16
(c) 2³ × 3² = 8 × 9 = 72

Answer: (a) 64 (b) 16 (c) 72

Multiplying and dividing powers

Write the powers out in full and count: 3⁴ × 3² = (3 × 3 × 3 × 3) × (3 × 3) = 3⁶. So when you multiply powers of the same base, add the indices. When you divide, cancel, which subtracts the indices.

Laws: am × aⁿ = am ⁺ ⁿ am ÷ aⁿ = am − ⁿ

Worked example: numbers

(a) 7⁵ × 7³ = 7⁵ ⁺ ³ = 7⁸ (b) 9¹⁰ ÷ 9⁴ = 9¹⁰ − ⁴ = 9⁶
(c) 2³ × 2⁴ ÷ 2⁵ = 2³ ⁺ ⁴ − ⁵ = 2² = 4

Answer: (a) 7⁸ (b) 9⁶ (c) 4

Worked example: letters and numbers

(a) x⁴ × x = x⁴ × x¹ = x⁵
(b) 3y⁵ × 4y²: multiply the numbers 3 × 4 = 12, add the indices: 12y⁷
(c) 20m⁸ ÷ 5m³: divide the numbers 20 ÷ 5 = 4, subtract the indices: 4m⁵

Answer: (a) x⁵ (b) 12y⁷ (c) 4m⁵

Power of a power and the zero index

(5²)³ = 5² × 5² × 5² = 5⁶. So a power of a power multiplies the indices.

Any number divided by itself is 1. 6³ ÷ 6³ = 6³ − ³ = 6⁰, so 6⁰ = 1.

Laws: (am)ⁿ = am × ⁿ a⁰ = 1 (for any a that is not 0)

Worked example: power of a power

(a) (4³)² = 4⁶ (b) (x⁵)⁴ = x²⁰ (c) (2y³)² = 2² × y⁶ = 4y⁶
(d) 8⁰ + 3⁰ = 1 + 1 = 2

Answer: (a) 4⁶ (b) x²⁰ (c) 4y⁶ (d) 2

Negative indices

Follow the pattern down: 2³ = 8, 2² = 4, 2¹ = 2, 2⁰ = 1, 2−¹ = 1/2, 2−² = 1/4. Each step divides by 2. A negative power means “one over”.

• 1000; 10² 100; 10¹ 10; 10⁰ 1; 10−¹ 1/10 = 0.1; 10−² 1/100 = 0.01; 10−³ 1/1000 = 0.001

Law: a−ⁿ = 1/aⁿ

Worked example: negative powers

(a) 5−² = 1/5² = 1/25 (b) 4−¹ = 1/4 (c) 10−³ = 1/1000 = 0.001
(d) 3⁴ ÷ 3⁶ = 3⁴ − ⁶ = 3−² = 1/9

Answer: (a) 1/25 (b) 1/4 (c) 0.001 (d) 1/9

Common mistakes

• Multiplying the base numbers: 3² × 3⁴ is 3⁶, not 9⁶.
• Multiplying the indices when you should add them: x³ × x⁴ = x⁷, not x¹².
• Thinking a⁰ = 0. Any non-zero number to the power 0 is 1.
• Thinking a negative power gives a negative number: 2−³ = 1/8, not −8.
• Using the laws with different bases: 2³ × 5² cannot be written as one power.

Revision questions

Try these, then check your answers below each one.

Question 1. Write 5³ × 5⁴ as a single power of 5. (Choose from: 5⁷, 5¹², 25⁷, 25¹².)

Answer: 5⁷

Working: 5³⁺⁴ = 5⁷.

Question 2. Simplify x⁹ ÷ x³.

Answer: x^6

Working: x⁹⁻³ = x⁶.

Question 3. Work out the value of 7⁰.

Answer: 1

Working: 7⁰ = 1.

Related lessons

• Previous lesson: 18.1 Multiplying and Dividing Fractions
• Next lesson: 18.3 Writing Large Numbers in Standard Form
• In this chapter: 18.4 Writing Small Numbers in Standard Form
• In this chapter: 18.5 Calculating with Standard Form

Continue in the full course

This lesson is section 18.2 (EGCMF1802) 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.