Exponents and Powers
1. Introduction
An exponent (or power) tells us how many times a number (the BASE) is multiplied by itself.
aⁿ = a × a × a × ... (n times)
a is the BASE, n is the EXPONENT, and aⁿ is the POWER.
Example: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. Base = 2, Exponent = 5.
'Exponents give us a SHORTHAND way to write repeated multiplication.'
2. Laws of Exponents (for Positive Integers)
| Law | Formula | Example |
|---|---|---|
| Product of powers | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ = 128 |
| Quotient of powers | aᵐ ÷ aⁿ = aᵐ⁻ⁿ (m > n) | 5⁶ ÷ 5² = 5⁴ = 625 |
| Power of a power | (aᵐ)ⁿ = aᵐⁿ | (3²)⁴ = 3⁸ = 6561 |
| Power of a product | (ab)ᵐ = aᵐ × bᵐ | (2×3)⁴ = 2⁴ × 3⁴ = 16 × 81 = 1296 |
| Power of a quotient | (a/b)ᵐ = aᵐ/bᵐ | (2/3)³ = 2³/3³ = 8/27 |
3. Zero and Negative Exponents
Zero Exponent
a⁰ = 1 for any non-zero a.
Proof: aᵐ ÷ aᵐ = aᵐ⁻ᵐ = a⁰, but also aᵐ ÷ aᵐ = 1, so a⁰ = 1.
Negative Exponents
a⁻ⁿ = 1/aⁿ for any non-zero a.
Example: 2⁻³ = 1/2³ = 1/8.
Worked Example: Simplify and express with positive exponents: (3⁻⁴ × 3²) ÷ 3⁻¹
3⁻⁴ × 3² = 3⁽⁻⁴⁺²⁾ = 3⁻² 3⁻² ÷ 3⁻¹ = 3⁽⁻²⁻⁽⁻¹⁾⁾ = 3⁻¹ = 1/3
4. Laws Extended to All Integers
All the laws work for NEGATIVE exponents too:
- aᵐ × aⁿ = aᵐ⁺ⁿ (including negative exponents)
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ (works for ALL integers)
- (aᵐ)ⁿ = aᵐⁿ (works for ALL integers)
Worked Example: Simplify (2⁻³ × 2⁵) ÷ 2⁻²
2⁻³ × 2⁵ = 2² 2² ÷ 2⁻² = 2²⁻⁽⁻²⁾ = 2⁴ = 16
5. Standard Form (Scientific Notation)
Writing numbers as A × 10ⁿ where 1 ≤ A < 10 and n is an integer.
| Type | Number | Standard Form |
|---|---|---|
| Very LARGE | 3,00,00,000 m/s (speed of light) | 3.0 × 10⁸ m/s |
| Very LARGE | 1,50,000,000 km (Earth-Sun distance) | 1.5 × 10⁸ km |
| Very SMALL | 0.000000001 m (atom diameter) | 1.0 × 10⁻⁹ m |
| Very SMALL | 0.00000000000000000016 C (electron charge) | 1.6 × 10⁻¹⁹ C |
Converting to Standard Form
Large numbers: Move decimal LEFT. n = number of places moved (positive). Example: 450000 = 4.5 × 10⁵
Small numbers: Move decimal RIGHT. n = number of places moved (negative). Example: 0.00078 = 7.8 × 10⁻⁴
6. Comparing Numbers in Standard Form
First compare the exponent (n). The LARGER the exponent, the LARGER the number.
| Number | Comparison |
|---|---|
| 4.2 × 10⁶ vs 8.1 × 10⁵ | 10⁶ > 10⁵, so 4.2 × 10⁶ > 8.1 × 10⁵ |
| 3.7 × 10⁻³ vs 9.2 × 10⁻⁴ | 10⁻³ > 10⁻⁴, so 3.7 × 10⁻³ > 9.2 × 10⁻⁴ |
| 5.2 × 10⁵ vs 5.2 × 10⁴ | 5.2 × 10⁵ > 5.2 × 10⁴ |
| 2.8 × 10⁶ vs 9.1 × 10⁶ | Same exponent: 2.8 × 10⁶ < 9.1 × 10⁶ |
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| 'a⁰ = 0' | a⁰ = 1 for any non-zero a |
| '2a³ = (2a)³' | 2a³ = 2 × a × a × a. (2a)³ = 8a³ |
| 'a⁻ⁿ is negative' | a⁻ⁿ = 1/aⁿ — it is the RECIPROCAL, not necessarily NEGATIVE |
| 'Adding exponents when bases differ' | a² × b³ cannot be simplified by adding exponents |
| 'aᵐ × aⁿ = aᵐⁿ' | WRONG! aᵐ × aⁿ = aᵐ⁺ⁿ (ADD, not multiply) |
ICSE Exam Focus (4–6 marks)
- 2-mark questions: Simplify using laws of exponents
- 3-mark questions: Express in standard form or evaluate with negative exponents
- 4-mark questions: Word problems comparing numbers in scientific notation
- 6-mark questions: Combined operations with powers and standard form
Self-Test
Q1. Simplify: (2⁻³ × x⁻⁵) × (2⁵ × x³) A1. 2⁽⁻³⁺⁵⁾ × x⁽⁻⁵⁺³⁾ = 2² × x⁻² = 4/x²
Q2. Express 0.00000000712 in standard form. A2. 7.12 × 10⁻⁹
Q3. Evaluate: (4⁻¹ + 2⁻²)⁻¹ A3. 4⁻¹ = 1/4, 2⁻² = 1/4. Sum = 1/2. Then (1/2)⁻¹ = 2.
Q4. The mass of Earth is 5.97 × 10²⁴ kg and the mass of Mars is 6.42 × 10²³ kg. Which is heavier and by how many times? A4. Earth is heavier. Ratio = 5.97 × 10²⁴ / 6.42 × 10²³ = 5.97/6.42 × 10 ≈ 0.93 × 10 = 9.3. Earth is about 9.3 times heavier.
Q5. Simplify and write with positive exponent: (a²b⁻³)⁴ × (a⁻¹b)² A5. a⁸b⁻¹² × a⁻²b² = a⁶b⁻¹⁰ = a⁶/b¹⁰
Q6. Express 3.2 × 10⁵ and 4.5 × 10⁴ in usual form. Compare them. A6. 3.2 × 10⁵ = 320,000. 4.5 × 10⁴ = 45,000. 320,000 > 45,000.
