Mastering Integers, Indices, and Scientific Notation

1. Operations with Integers

Addition Rules

• Same signs: Add the numbers and keep the original sign.
  Example: -4 + (-6) = -10
• Different signs: Subtract the smaller number from the larger number and keep the sign of the larger number.
  Example: 7 + (-3) = 4

Subtraction Rules

To subtract, turn the operation into addition by changing the sign of the second number:

a - b = a + (-b)

Step-by-Step Example:
Solve: 5 - 8
Step 1: Rewrite as addition → 5 + (-8)
Step 2: Different signs, so subtract → 8 - 5 = 3
Step 3: Keep the sign of the larger number → -3

Multiplication and Division Sign Rules

Step-by-Step Example:
Solve: -3 × -4
Step 1: Same signs result in a positive product.
Step 2: 3 × 4 = 12
Answer: 12

2. Indices and Powers

An index (or power) represents repeated multiplication: aⁿ = a × a × a... (n times)

Example: 2³ = 2 × 2 × 2 = 8

3. Fundamental Laws of Indices

Multiplication Law

aᵐ × aⁿ = aᵐ⁺ⁿ

Worked Example: 2³ × 2⁴

• Step 1: Since the bases are the same, add the powers.
• Step 2: 2³⁺⁴ = 2⁷
• Step 3: 2⁷ = 128

Division Law

aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Worked Example: 5⁶ ÷ 5²

• Step 1: Since the bases are the same, subtract the powers.
• Step 2: 5⁶⁻² = 5⁴
• Step 3: 5⁴ = 625

Power of a Power Law

(aᵐ)ⁿ = aᵐˣⁿ

Worked Example: (3²)⁴

• Step 1: Multiply the powers → 2 × 4 = 8
• Step 2: 3⁸ = 6561

Power of a Product Law

(ab)ⁿ = aⁿ × bⁿ

Worked Example: (2 × 3)²

• Step 1: Apply the power to both numbers inside the parentheses.
• Step 2: 2² × 3² = 4 × 9
• Answer: 36

4. Negative and Zero Powers

Negative Powers

a⁻ⁿ = 1 / aⁿ

Worked Example: 5⁻²

• Step 1: Flip the base to create a reciprocal → 1 / 5²
• Step 2: Calculate the power → 5² = 25
• Answer: 1/25

Zero Power

Any non-zero base raised to the power of zero is equal to one: a⁰ = 1

Example: 7⁰ = 1

5. Scientific Notation Fundamentals

Standard Form:

a × 10ⁿ (where 1 ≤ a < 10)

Converting Numbers

Large Numbers (Positive Power)

Worked Example: Convert 4500

• Step 1: Move the decimal point to create a number between 1 and 10 → 4.5
• Step 2: Count the places moved (3 places) → 10³
• Answer: 4.5 × 10³

Small Numbers (Negative Power)

Worked Example: Convert 0.0045

• Step 1: Move the decimal point to create a number between 1 and 10 → 4.5
• Step 2: Count the places moved (3 places to the right) → 10⁻³
• Answer: 4.5 × 10⁻³

6. Operations with Scientific Notation

Multiplication

(2 × 10³) × (3 × 10⁴)

• Step 1: Multiply the coefficients → 2 × 3 = 6
• Step 2: Add the powers of ten → 10³⁺⁴ = 10⁷
• Answer: 6 × 10⁷

Division

(6 × 10⁵) ÷ (2 × 10²)

• Step 1: Divide the coefficients → 6 ÷ 2 = 3
• Step 2: Subtract the powers of ten → 10⁵⁻² = 10³
• Answer: 3 × 10³