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Mathematics · Chapter 2: Real Numbers
Laws of Indices
Product, quotient and power laws, zero, negative and fractional indices, with worked examples.
Key termRememberExampleWatch out
01Concept
Writing 2 × 2 × 2 × 2 × 2 is tiring. We write it as 2⁵. Here 2 is the base and 5 is the index (or power, or exponent): the index tells you how many times the base is multiplied by itself.
The laws
| Law | Rule | Example |
|---|---|---|
| Product law | aᵐ × aⁿ = aᵐ⁺ⁿ | 3² × 3⁴ = 3⁶ |
| Quotient law | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁷ ÷ 5³ = 5⁴ |
| Power law | (aᵐ)ⁿ = aᵐⁿ | (2³)² = 2⁶ = 64 |
| Power of a product | (ab)ⁿ = aⁿbⁿ | (2x)³ = 8x³ |
| Power of a quotient | (a/b)ⁿ = aⁿ/bⁿ | (2/3)² = 4/9 |
| Zero index | a⁰ = 1 (a ≠ 0) | 7⁰ = 1, (−5)⁰ = 1 |
| Negative index | a⁻ⁿ = 1/aⁿ | 2⁻³ = 1/8 |
| Fractional index | a^(1/n) = ⁿ√a and a^(m/n) = (ⁿ√a)ᵐ | 8^(1/3) = 2; 27^(2/3) = 3² = 9 |
Why a⁰ = 1
By the quotient law, 5³ ÷ 5³ = 5⁰. But any number divided by itself is 1. So 5⁰ must equal 1.
Working with fractional indices
Do the root first, then the power: the numbers stay small. 16^(3/4) = (⁴√16)³ = 2³ = 8.
A negative fractional index means root, power, then turn upside down. 32^(−2/5) = 1/(⁵√32)² = 1/2² = 1/4.
02Virtual Lab
Try this topic for yourself in the virtual lab. It opens on its own page with steps, tasks and questions.
03Key Formulas
Product law
Quotient law
Power law
Zero and negative index
a ≠ 0
Fractional index
04Real World Examples
Computer memory
1 KB = 2¹⁰ = 1,024 bytes, 1 MB = 2²⁰ bytes. A 64 GB phone stores 64 × 2³⁰ bytes.
Spreading a message
If you forward a WhatsApp message to 3 friends, and each forwards it to 3 more, after 5 rounds it reaches 3⁵ = 243 people.
Area and volume
A square plot of side 30 m has an area of 30² = 900 m²; a cubic water tank of side 2 m holds 2³ = 8 m³.
05Practice Questions
Try to answer in your own words first, then open the model answer.
Short questions
Q1.Simplify x⁵ × x³ ÷ x².Show answer
Answer
x⁵⁺³⁻² = x⁶.
Q2.Evaluate 4⁻² and (2/5)⁻¹.Show answer
Answer
4⁻² = 1/16; (2/5)⁻¹ = 5/2.
Q3.Evaluate 125^(2/3).Show answer
Answer
(∛125)² = 5² = 25.
Q4.Simplify (3a²b)³.Show answer
Answer
3³a⁶b³ = 27a⁶b³.
Q5.Solve 3ˣ = 81.Show answer
Answer
81 = 3⁴, so x = 4.
Q6.Simplify (2⁵ × 2⁻³)/2⁰.Show answer
Answer
2⁵⁻³ ÷ 1 = 2² = 4.
Long questions
Q7.Simplify (16x⁸)^(1/4) × x⁻¹ and explain each law used.Show answer
Answer
- (16x⁸)^(1/4) = 16^(1/4) × (x⁸)^(1/4) (power of a product)
- = 2 × x² (fourth root of 16 is 2; power law: 8 × ¼ = 2)
- 2x² × x⁻¹ = 2x²⁻¹ (product law)
- = 2x