Icse Laws Exponents
Number Systems — Icse Laws Exponents
Icse Laws Exponents
Laws of Exponents
What are Exponents?
aⁿ means "a multiplied by itself n times" — a is the base, n is the exponent (or index or power).
2⁴ = 2 × 2 × 2 × 2 = 16
The Seven Laws
| Law | Statement | Example |
|---|---|---|
| L1 — Product | aᵐ × aⁿ = aᵐ⁺ⁿ | 2³ × 2⁴ = 2⁷ |
| L2 — Quotient | aᵐ ÷ aⁿ = aᵐ⁻ⁿ | 5⁶ ÷ 5² = 5⁴ |
| L3 — Power of a Power | (aᵐ)ⁿ = aᵐⁿ | (3²)⁴ = 3⁸ |
| L4 — Power of a Product | (ab)ⁿ = aⁿ × bⁿ | (2×3)⁴ = 2⁴ × 3⁴ |
| L5 — Power of a Quotient | (a/b)ⁿ = aⁿ/bⁿ | (4/5)³ = 4³/5³ |
| L6 — Zero Exponent | a⁰ = 1 (a ≠ 0) | 7⁰ = 1 |
| L7 — Negative Exponent | a⁻ⁿ = 1/aⁿ | 3⁻² = 1/9 |
Worked Examples
Example 1 (L1 + L2):
(2³ × 2⁴) ÷ 2⁵ = 2⁷ ÷ 2⁵ = 2² = 4
Example 2 (L3):
(x³)⁴ = x¹² | (2²)³ = 2⁶ = 64
Example 3 (L4 + L7):
(2a)⁻³ = 2⁻³ × a⁻³ = a⁻³/8
Example 4 (mixed):
(3² × 3⁻⁵) × 3⁴ = 3²⁻⁵⁺⁴ = 3¹ = 3
Example 5 (negative exponents):
Simplify: (2⁻² × 3²) / (4⁻¹ × 6²) = (1/4 × 9) / (1/4 × 36) = (9/4) / (36/4) = 9/36 = 1/4
Standard Form (Scientific Notation)
A number in standard form = a × 10ⁿ where 1 ≤ a < 10, n is an integer.
| Number | Standard Form |
|---|---|
| 3,500,000 | 3.5 × 10⁶ |
| 0.00042 | 4.2 × 10⁻⁴ |
| 602,000,000,000,000,000,000,000 | 6.02 × 10²³ |
Multiplying in standard form:
(3 × 10⁴) × (2 × 10³) = 6 × 10⁷
Dividing in standard form:
(8 × 10⁶) ÷ (4 × 10²) = 2 × 10⁴
Common Mistakes
| Error | Wrong | Correct |
|---|---|---|
| Adding bases | 2³ × 3³ = 6⁶ | (2×3)³ = 6³ (L4) |
| Multiplying exponents when multiplying | aᵐ × aⁿ = aᵐⁿ | aᵐ × aⁿ = aᵐ⁺ⁿ (L1) |
| Zero power confusion | 0⁰ = 0 | 0⁰ is undefined; a⁰ = 1 for a ≠ 0 |
| Negative exponent = negative | 2⁻³ = −8 | 2⁻³ = 1/8 |
ICSE Standard Question Types
-
Simplify: (x²y³)⁴ / (xy)⁵ → x⁸y¹²/x⁵y⁵ = x³y⁷
-
Find x: 3ˣ × 3² = 3⁷ → x + 2 = 7 → x = 5
-
Evaluate: (2⁻¹ + 3⁻¹) × 6² → (1/2 + 1/3) × 36 = (5/6) × 36 = 30
Quick Check
- Simplify: a⁵ × a⁻³ × a²
- Find x: 5ˣ⁺¹ = 5³
- Write 0.000045 in standard form.
- Evaluate: (3²)³ ÷ 3⁵
- Stretch: Prove that (aᵐ)ⁿ = (aⁿ)ᵐ using the product law.
Key Takeaways (TL;DR)
- What are Exponents?
- The Seven Laws
- Worked Examples
- Standard Form (Scientific Notation)
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