Subject: Mathematics | Level: JHS (Basic 7 – 9) | Topic: Indices and Standard Form | Difficulty: Intermediate to Advanced
Introduction
Indices (also called exponents or powers) provide a shorthand way of writing repeated multiplication. Instead of writing 3 × 3 × 3 × 3 × 3, we write 3⁵ (read as “three to the power of five”). Indices appear throughout Mathematics and Science, from very large numbers like the distance from the Earth to the Sun, to very small ones like the size of a bacterium. Understanding and applying the laws of indices correctly is essential for BECE success.
1. Basic Notation
In the expression aⁿ:
- a is the base
- n is the index (exponent or power)
- aⁿ means “a multiplied by itself n times”
Examples: 2⁴ = 2 × 2 × 2 × 2 = 16 5³ = 5 × 5 × 5 = 125 10² = 100
2. The Laws of Indices
These rules apply only when the bases are the same:
Law 1: Multiplication (Add the powers)
aᵐ × aⁿ = aᵐ⁺ⁿ
Example: 3⁴ × 3² = 3⁴⁺² = 3⁶
Example: x³ × x⁵ = x⁸
Law 2: Division (Subtract the powers)
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Example: 5⁷ ÷ 5³ = 5⁷⁻³ = 5⁴
Example: y⁶ ÷ y² = y⁴
Law 3: Power of a Power (Multiply the powers)
(aᵐ)ⁿ = aᵐˣⁿ
Example: (2³)⁴ = 2¹²
Example: (x²)⁵ = x¹⁰
Law 4: Zero Index
a⁰ = 1 (for any non-zero value of a)
Example: 7⁰ = 1 100⁰ = 1 x⁰ = 1
Law 5: Negative Index (Reciprocal)
a⁻ⁿ = 1/aⁿ
Example: 2⁻³ = 1/2³ = 1/8
Example: 5⁻² = 1/25
Example: x⁻¹ = 1/x
Law 6: Fractional Index (Roots)
a^(1/n) = nth root of a
Example: 25^(1/2) = √25 = 5
Example: 8^(1/3) = ∛8 = 2 (cube root of 8)
Example: 16^(3/4) = (16^(1/4))³ = (∜16)³ = 2³ = 8
3. Worked Examples Using Multiple Laws
Example 1: Simplify: a⁵ × a³ ÷ a⁴
= a⁵⁺³ ÷ a⁴
= a⁸ ÷ a⁴
= a⁸⁻⁴ = a⁴
Example 2: Evaluate: (2³ × 2⁴) ÷ 2⁵
= 2⁷ ÷ 2⁵ = 2² = 4
Example 3: Simplify: (3x²)³
= 3³ × x²ˣ³ = 27x⁶
Example 4: Evaluate: 27^(2/3)
= (27^(1/3))² = (∛27)² = 3² = 9
Example 5: Evaluate: 4⁻² + 2⁰
= 1/16 + 1 = 17/16
4. Standard Form (Scientific Notation)
Standard form is a way of writing very large or very small numbers in a compact, manageable way.
A number in standard form is written as: A × 10ⁿ
where: 1 ≤ A < 10 and n is a positive or negative integer
Writing Numbers in Standard Form
Large numbers (positive index):
- 5,700,000 = 5.7 × 10⁶ (move decimal 6 places left)
- 34,500 = 3.45 × 10⁴
- 820,000,000 = 8.2 × 10⁸
Small numbers (negative index):
- 0.00034 = 3.4 × 10⁻⁴ (move decimal 4 places right)
- 0.0000072 = 7.2 × 10⁻⁶
Converting from Standard Form to Ordinary Numbers
- 6.3 × 10⁵ = 630,000
- 2.15 × 10³ = 2,150
- 4.8 × 10⁻³ = 0.0048
Calculations with Standard Form
Example: Calculate (3 × 10⁴) × (2 × 10³)
= (3 × 2) × (10⁴ × 10³)
= 6 × 10⁷
Example: Calculate (8 × 10⁶) ÷ (4 × 10²)
= (8 ÷ 4) × (10⁶ ÷ 10²)
= 2 × 10⁴
Example: Write in standard form: the population of Ghana is approximately 33,500,000.
= 3.35 × 10⁷
Summary
- aᵐ × aⁿ = aᵐ⁺ⁿ (multiply: add powers)
- aᵐ ÷ aⁿ = aᵐ⁻ⁿ (divide: subtract powers)
- (aᵐ)ⁿ = aᵐⁿ (power of a power: multiply)
- a⁰ = 1 (zero index = 1)
- a⁻ⁿ = 1/aⁿ (negative index = reciprocal)
- a^(1/n) = nth root of a
- Standard form: A × 10ⁿ where 1 ≤ A < 10
Practice Questions
- Simplify: x⁴ × x³ ÷ x²
- Evaluate: 3⁻² + 3⁰
- Find the value of: 64^(2/3)
- Write in standard form: (a) 4,720,000 (b) 0.00000583
- Calculate: (5 × 10³) × (4 × 10⁴). Give your answer in standard form.
Answers: (1) x⁵ (2) 1/9 + 1 = 10/9 (3) (∛64)² = 4² = 16 (4a) 4.72 × 10⁶ (4b) 5.83 × 10⁻⁶ (5) 20 × 10⁷ = 2 × 10⁸
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