Indices – Basic 6 Mathematics Lesson Note (Second Term)

What are Indices?

An index (or power) tells us how many times to multiply a number by itself.

Examples:

• 2³ = 2 × 2 × 2 = 8
• 5² = 5 × 5 = 25
• 10⁴ = 10 × 10 × 10 × 10 = 10,000

Parts of Index Notation:

• Base: The number being multiplied (2 in 2³)
• Index/Power: How many times to multiply (3 in 2³)

Reading Index Numbers

Examples:

• 3² = “3 to the power of 2” or “3 squared”
• 4³ = “4 to the power of 3” or “4 cubed”
• 5⁴ = “5 to the power of 4”

Numbers in Index Form

Writing Numbers in Index Form

Example 1: Express 8 as a power of 2 8 = 2 × 2 × 2 = 2³

Example 2: Express 125 as a power of 5 125 = 5 × 5 × 5 = 5³

Example 3: Express 1000 as a power of 10 1000 = 10 × 10 × 10 = 10³

Exercise A – Writing in Index Form

Write these numbers in index form:

1. 16 as a power of 2 = ______
2. 27 as a power of 3 = ______
3. 81 as a power of 3 = ______
4. 64 as a power of 4 = ______
5. 32 as a power of 2 = ______
6. 100 as a power of 10 = ______
7. 49 as a power of 7 = ______
8. 243 as a power of 3 = ______

Finding the Value of Index Numbers

Calculate these:

• 3⁴ = 3 × 3 × 3 × 3 = 81
• 2⁵ = 2 × 2 × 2 × 2 × 2 = 32
• 5³ = 5 × 5 × 5 = 125

Exercise B – Finding Values

Calculate the value of these index numbers:

1. 2⁴ = ______
2. 3³ = ______
3. 5² = ______
4. 4³ = ______
5. 6² = ______
6. 2⁶ = ______
7. 10³ = ______
8. 7² = ______
9. 8² = ______
10. 9² = ______

Rules of Indices

Rule 1: Multiplication (Same Base)

When multiplying powers with the same base, add the indices

Formula: aᵐ × aⁿ = aᵐ⁺ⁿ

Examples:

• 2³ × 2² = 2³⁺² = 2⁵ = 32
• 5² × 5⁴ = 5²⁺⁴ = 5⁶ = 15,625

Rule 2: Division (Same Base)

When dividing powers with the same base, subtract the indices

Formula: aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Examples:

• 3⁵ ÷ 3² = 3⁵⁻² = 3³ = 27
• 10⁶ ÷ 10⁴ = 10⁶⁻⁴ = 10² = 100

Rule 3: Power of a Power

When raising a power to another power, multiply the indices

Formula: (aᵐ)ⁿ = aᵐˣⁿ

Examples:

• (2³)² = 2³ˣ² = 2⁶ = 64
• (5²)³ = 5²ˣ³ = 5⁶ = 15,625

Rule 4: Any Number to Power 0

Any number (except 0) raised to power 0 equals 1

Formula: a⁰ = 1 (where a ≠ 0)

Examples:

• 5⁰ = 1
• 100⁰ = 1
• 2,340⁰ = 1

Rule 5: Any Number to Power 1

Any number raised to power 1 equals itself

Formula: a¹ = a

Examples:

• 7¹ = 7
• 25¹ = 25

Exercise C – Using Index Rules

Simplify using index rules:

1. 3² × 3³ = ______
2. 5⁴ ÷ 5² = ______
3. (2³)² = ______
4. 4⁰ = ______
5. 6¹ = ______
6. 2⁵ × 2¹ = ______
7. 10⁶ ÷ 10³ = ______
8. (3²)³ = ______
9. 7³ × 7² = ______
10. 8⁴ ÷ 8⁴ = ______

Exercise D – Mixed Index Operations

Solve these problems:

1. 2³ × 2² × 2¹ = ______
2. 5⁵ ÷ 5³ ÷ 5¹ = ______
3. (4²)² × 4¹ = ______
4. 3⁴ × 3⁰ = ______
5. (2³ × 2²) ÷ 2⁴ = ______

Real Life Problems

Population Growth

Example: A town’s population doubles every 10 years. If the current population is 5,000, what will it be in 30 years?

Solution:

• After 10 years: 5,000 × 2¹ = 10,000
• After 20 years: 5,000 × 2² = 20,000
• After 30 years: 5,000 × 2³ = 40,000

Exercise E – Population and Growth

Solve these growth problems:

1. Bacteria Growth: Bacteria triple every hour. If there are initially 100 bacteria, how many will there be after 4 hours? Answer: ______ bacteria
2. School Enrollment: A school’s enrollment doubles every 5 years. If there are currently 200 students, how many will there be in 15 years? Answer: ______ students
3. Savings Account: Money in a savings account increases by 50% each year. If Kemi starts with ₦1,000, how much will she have after 3 years? (Use 1.5 as the multiplier) Answer: ₦______

Exercise F – Area and Volume

Solve these measurement problems:

1. Square Field: A square field has sides of 6 meters. What is its area? (Area = side²) Answer: ______ square meters
2. Cube Storage: A cube-shaped container has sides of 4 meters. What is its volume? (Volume = side³) Answer: ______ cubic meters
3. Computer Memory: Computer memory is often measured in powers of 2. If each unit stores 2⁸ bytes, how many bytes can 4 units store? Answer: ______ bytes

Exercise G – Technology and Science

Apply indices in technology:

1. Digital Storage: A computer has 2¹⁰ GB of storage. How much storage is this in actual numbers? Answer: ______ GB
2. Network Speed: Internet speed doubles every year. If current speed is 10 Mbps, what will it be in 4 years? Answer: ______ Mbps
3. Solar Panel Output: A solar panel’s efficiency improves by a factor of 1.2 each year. If it currently produces 100 watts, what will it produce in 3 years? Answer: ______ watts (round to nearest whole number)

Quantitative Reasoning

Pattern Recognition

Example: Find the pattern: 2, 4, 8, 16, 32, … Pattern: Each term is 2 raised to increasing powers 2¹, 2², 2³, 2⁴, 2⁵, …

Exercise H – Number Patterns

Complete these patterns:

1. Powers of 3: 3, 9, 27, _____, _____
2. Powers of 5: 5, 25, 125, _____, _____
3. Powers of 2: 2, 4, 8, 16, _____, _____
4. Powers of 10: 10, 100, 1000, _____, _____

Exercise I – Comparing Powers

Compare these using >, <, or =:

1. 2⁵ _____ 3³
2. 4² _____ 2⁴
3. 5² _____ 3³
4. 10² _____ 2⁵
5. 6² _____ 4³

Exercise J – Problem Solving

Use logical thinking:

1. Missing Exponent: 2ˣ = 64. What is x? Answer: x = ______
2. Missing Base: x³ = 216. What is x? Answer: x = ______
3. Equation Solving: If 3ˣ × 3² = 3⁵, what is x? Answer: x = ______
4. Growth Calculation: A plant’s height doubles each week. If it’s currently 8 cm tall, when was it 1 cm tall? Answer: ______ weeks ago

Answer Key

Exercise A:

1. 2⁴, 2. 3³, 3. 3⁴, 4. 4³, 5. 2⁵, 6. 10², 7. 7², 8. 3⁵

Exercise B:

1. 16, 2. 27, 3. 25, 4. 64, 5. 36, 6. 64, 7. 1000, 8. 49, 9. 64, 10. 81

Exercise C:

1. 3⁵, 2. 5², 3. 2⁶, 4. 1, 5. 6, 6. 2⁶, 7. 10³, 8. 3⁶, 9. 7⁵, 10. 1

Exercise D:

1. 2⁶, 2. 5¹, 3. 4⁵, 4. 3⁴, 5. 2¹

Exercise E:

1. 8,100 bacteria
2. 1,600 students
3. ₦3,375

Exercise F:

1. 36 square meters
2. 64 cubic meters
3. 1,024 bytes

Exercise G:

1. 1,024 GB
2. 160 Mbps
3. 173 watts

Exercise H:

1. 81, 243
2. 625, 3,125
3. 32, 64
4. 10,000, 100,000

Exercise I:

1. (32 > 27)
    
2. = (16 = 16)
3. < (25 < 27)
4. (100 > 32)
    
5. < (36 < 64)

Exercise J:

1. x = 6
2. x = 6
3. x = 3
4. 3 weeks ago