Exponents
1. What Is an Exponent?
An EXPONENT tells how many times a number (the BASE) is multiplied by itself.
aⁿ = a × a × a × ... (n times)
- a = BASE (the number being multiplied)
- n = EXPONENT or POWER (how many times to multiply)
- aⁿ = 'a raised to the power n'
Examples:
- 2⁵ = 2 × 2 × 2 × 2 × 2 = 32
- (-3)⁴ = (-3) × (-3) × (-3) × (-3) = 81
- (-2)³ = (-2) × (-2) × (-2) = -8
Reading Exponents
- a² = 'a squared'
- a³ = 'a cubed'
- a⁴ = 'a raised to power 4'
2. Laws of Exponents (for Same Base)
| Law | Formula | Example |
|---|---|---|
| Product | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ |
| Quotient | aᵐ ÷ aⁿ = aᵐ⁻ⁿ (m > n, a ≠ 0) | 2⁵ ÷ 2² = 2³ |
| Power of Power | (aᵐ)ⁿ = aᵐⁿ | (2³)² = 2⁶ |
| Power of Product | (ab)ᵐ = aᵐ bᵐ | (2×3)² = 2² × 3² |
| Power of Quotient | (a/b)ᵐ = aᵐ / bᵐ (b ≠ 0) | (2/3)² = 2²/3² |
| Zero Exponent | a⁰ = 1 (a ≠ 0) | 5⁰ = 1 |
| Negative Exponent | a⁻ⁿ = 1/aⁿ (a ≠ 0) | 2⁻³ = 1/2³ = 1/8 |
Important Notes
- The laws apply ONLY when bases are the SAME (for product and quotient rules).
- 0⁰ is NOT defined.
- 1ⁿ = 1 for any value of n.
3. Negative Exponents
A negative exponent means RECIPROCAL. a⁻ⁿ = 1/aⁿ
Worked Examples (ICSE 2024, 2 marks)
Simplify and express with positive exponent: (3⁻⁴ × 3⁵) ÷ 3⁻².
Solution: 3⁻⁴ × 3⁵ = 3¹ (using product rule: -4 + 5 = 1) 3¹ ÷ 3⁻² = 3¹⁻⁽⁻²⁾ = 3³ = 27.
Converting Between Forms
- 10⁻³ = 1/10³ = 1/1000 = 0.001
- 1/5² = 5⁻²
- (2/3)⁻¹ = 3/2 (reciprocal of the fraction)
Common Mistake
'a⁻ⁿ is NEGATIVE' — FALSE. a⁻ⁿ is the RECIPROCAL, which can be positive or negative depending on the base. Example: (-2)⁻³ = -1/8 (negative because base is negative and exponent is odd).
4. Zero Exponent
Any non-zero number raised to power ZERO equals 1.
Why Does a⁰ = 1?
By quotient rule: aᵐ ÷ aᵐ = aᵐ⁻ᵐ = a⁰. But aᵐ ÷ aᵐ = 1. Therefore a⁰ = 1.
Examples: 7⁰ = 1, (-100)⁰ = 1, (3/5)⁰ = 1.
5. Scientific Notation
Scientific notation expresses numbers as: a × 10ⁿ where 1 ≤ a < 10 and n is an integer.
Converting to Scientific Notation
Large numbers (positive exponent):
- 3,50,00,000 = 3.5 × 10⁷ (move decimal 7 places left)
- 93,000,000 = 9.3 × 10⁷
Small numbers (negative exponent):
- 0.000007 = 7.0 × 10⁻⁶ (move decimal 6 places right)
- 0.00000054 = 5.4 × 10⁻⁷
Worked Example (ICSE 2023, 3 marks)
'The distance from Earth to Sun is 149,600,000 km. Write in scientific notation.'
Solution: 1.496 × 10⁸ km.
Comparing Numbers in Scientific Notation
- Compare the exponent FIRST.
- Larger exponent = larger number.
- If exponents are equal, compare the coefficient (a).
Example: 2.5 × 10⁸ > 4.8 × 10⁷ (because 8 > 7).
6. Order of Magnitude
The ORDER OF MAGNITUDE is the power of 10 when a number is in scientific notation.
- 4.2 × 10⁶ → order of magnitude = 10⁶
- 7.89 × 10⁻⁴ → order of magnitude = 10⁻⁴
Rounding convention: If coefficient ≥ 5, round the exponent UP by 1.
- 8.1 × 10⁵ → order of magnitude = 10⁶ (since 8.1 ≥ 5)
- 2.3 × 10⁵ → order of magnitude = 10⁵ (since 2.3 < 5)
7. ICSE Exam Focus
| Topic | Marks | Frequency |
|---|---|---|
| Laws of exponents (simplify) | 3-4 marks | Very High |
| Negative exponents | 2-3 marks | High |
| Scientific notation | 2 marks | High |
| Zero exponent | 1 mark | Low |
Common Mistakes in ICSE Exams
- Adding exponents when multiplying DIFFERENT bases: 2³ × 3² ≠ 6⁵.
- Forgetting: a⁰ = 1, NOT 0.
- Writing a⁻ⁿ = -aⁿ (WRONG — it is 1/aⁿ).
- Not reducing to simplest exponential form.
Self-Test (5 Questions)
Q1. Simplify and express with positive exponent: (2⁻³ × 2⁵) ÷ 2⁻¹. (2 marks)
- A) 8
- B) 4
- C) 2
- D) 16
Q2. Evaluate: (3²)³ × 3⁻⁴. (2 marks)
Q3. Write in scientific notation: 0.00000092. (1 mark)
Q4. Simplify: (5/7)⁻² × (5/7)⁵. (2 marks)
- A) (5/7)³
- B) (7/5)³
- C) (5/7)⁷
- D) (7/5)⁷
Q5. Which is larger: 3.2 × 10⁵ or 8.9 × 10⁴? (1 mark)
Answers
A1. A) 8. (2⁻³⁺⁵ = 2². 2² ÷ 2⁻¹ = 2²⁻⁽⁻¹⁾ = 2³ = 8.) A2. 9. (3⁶ × 3⁻⁴ = 3² = 9.) A3. 9.2 × 10⁻⁷. A4. B) (7/5)³. ((5/7)⁻² = (7/5)². (7/5)² × (5/7)⁵. Or: (5/7)⁻²⁺⁵ = (5/7)³ = 1/(5/7)³ = (7/5)³.) A5. 3.2 × 10⁵ (since 10⁵ > 10⁴).
