Indices and Exponents
Introduction
Indices (also called exponents) provide a compact way to represent repeated multiplication. Understanding the laws of indices is essential for simplifying algebraic expressions and solving equations in ICSE Class 9.
Laws of Exponents
For positive integers m and n, and a ≠ 0:
| Law | Expression | Example |
|---|---|---|
| Product | a^m × a^n = a^(m+n) | x³ × x⁴ = x⁷ |
| Quotient | a^m ÷ a^n = a^(m-n), m > n | x⁷ ÷ x³ = x⁴ |
| Power of a power | (a^m)^n = a^(mn) | (x²)³ = x⁶ |
| Power of a product | (ab)^m = a^m × b^m | (2x)³ = 8x³ |
| Power of a quotient | (a/b)^m = a^m / b^m | (x/y)² = x²/y² |
Zero, Negative, and Fractional Exponents
Zero Exponent
a⁰ = 1 (for any a ≠ 0)
Example: 5⁰ = 1, (-3)⁰ = 1, x⁰ = 1
Negative Exponent
a^(-n) = 1/a^n (for a ≠ 0)
Example: 2^(-3) = 1/2³ = 1/8
Fractional Exponent
a^(m/n) = (ⁿ√a)^m = ⁿ√(a^m)
<h4>Example</h4> <p>8^(2/3) = (³√8)² = 2² = 4</p> <ICSEExample title="Simplify 27^(−2/3)"> <Solution> 27^(−2/3) = 1/27^(2/3) = 1/(³√27)² = 1/3² = 1/9 </Solution> </ICSEExample>Simplifying Expressions With Indices
<ICSEExample title="Simplify (2x³y⁻²)/(4x⁻¹y³)"> <Solution> (2x³y⁻²)/(4x⁻¹y³) = 2/4 × x³/x⁻¹ × y⁻²/y³ = 1/2 × x^(3-(-1)) × y^(-2-3) = 1/2 × x⁴ × y⁻⁵ = x⁴/(2y⁵) </Solution> </ICSEExample> <ICSEExample title="Simplify (a²b³)² × (a³b)³"> <Solution> (a²b³)² × (a³b)³ = a⁴b⁶ × a⁹b³ = a^(4+9) × b^(6+3) = a¹³b⁹ </Solution> </ICSEExample>Solving Exponential Equations
When bases are equal, equate the exponents.
<ICSEExample title="Solve 2^(x+1) = 32"> <Solution> 2^(x+1) = 32 2^(x+1) = 2⁵ x + 1 = 5 x = 4 </Solution> </ICSEExample>Common Mistakes With Fixes
| Mistake | Correction |
|---|---|
| Adding exponents when bases are different | Only add exponents when bases are the SAME |
| a^m × b^m ≠ (ab)^(2m) | (ab)^m = a^m × b^m (same exponent, not double) |
| Forgetting that a⁰ = 1 | Any non-zero number raised to 0 equals 1 |
| Confusing a^(-n) with -a^n | a^(-n) = 1/a^n, it is NOT -a^n |
ICSE Exam Focus
| Topic | Marks (approx.) | Frequency |
|---|---|---|
| Laws of indices | 3-4 marks | Very common |
| Simplification with negative/fractional exponents | 4-5 marks | Common |
| Solving exponential equations | 3 marks | Common |
| Evaluation of expressions | 3 marks | Frequently asked |
Self-Test
Q1: Simplify: (27x⁶y³)^(1/3)
Q2: Evaluate: 16^(−3/4)
Q3: Simplify: (3x²y⁻¹)³/(9x⁻¹y²)²
Q4: Solve: 3^(2x+1) = 81 (Answer: 81=3⁴. So 2x+1=4 → x=3/2.)
Q5: If 2^(x+3) = 8^(x−1), find x. (Answer: 8=2³. So 2^(x+3) = 2^(3x−3) → x+3=3x−3 → 2x=6 → x=3.)
ICSE Exam Tips
Indices questions in ICSE Class 9 typically appear in Section A (compulsory short-answer) for 2-3 marks and Section B for 4 marks. Common question patterns: (1) Simplify using laws. (2) Solve exponential equations by equating bases. (3) Evaluate numerically. 'Always express ALL terms with the SAME base before applying laws. For fractional exponents, remember a^(1/n) = ⁿ√a. For negative exponents, rewrite as 1/aⁿ before simplifying.'
Q5: Find the value of (0.00032)^(3/5)
