Answer: 6 × 10⁹

Example 2: (4 × 10³) × (5 × 10⁻²)

Solution: Step 1: Multiply: 4 × 5 = 20 Step 2: Add exponents: 10³⁺⁽⁻²⁾ = 10¹ Step 3: Combine: 20 × 10¹

Step 4: Adjust to proper form (1 ≤ a < 10): 20 × 10¹ = 2.0 × 10² = 2 × 10²

Answer: 2 × 10²

Example 3: (6.5 × 10⁻³) × (4 × 10⁷)

Solution: Multiply: 6.5 × 4 = 26 Add exponents: 10⁻³⁺⁷ = 10⁴ Combine: 26 × 10⁴

Adjust: 26 × 10⁴ = 2.6 × 10⁵

Answer: 2.6 × 10⁵

Dividing Scientific Notation

Rule: Divide the coefficients, subtract the exponents

(a × 10ᵐ) ÷ (b × 10ⁿ) = (a ÷ b) × 10ᵐ⁻ⁿ

Example 1: (8 × 10⁶) ÷ (2 × 10³)

Solution: Step 1: Divide coefficients: 8 ÷ 2 = 4 Step 2: Subtract exponents: 10⁶⁻³ = 10³ Step 3: Combine: 4 × 10³

Answer: 4 × 10³

Example 2: (9 × 10⁵) ÷ (3 × 10⁷)

Solution: Divide: 9 ÷ 3 = 3 Subtract exponents: 10⁵⁻⁷ = 10⁻² Combine: 3 × 10⁻²

Answer: 3 × 10⁻²

Example 3: (7.2 × 10⁴) ÷ (1.8 × 10⁻²)

Solution: Divide: 7.2 ÷ 1.8 = 4 Subtract exponents: 10⁴⁻⁽⁻²⁾ = 10⁴⁺² = 10⁶ Combine: 4 × 10⁶

Answer: 4 × 10⁶

Adding and Subtracting Scientific Notation

Rule: Exponents must be the SAME before adding/subtracting

Steps:

1. Convert numbers to have the same exponent
2. Add or subtract the coefficients
3. Keep the common exponent
4. Adjust to proper scientific notation if needed

Example 1: (3 × 10⁴) + (5 × 10⁴)

Solution: Same exponents! Just add coefficients: (3 + 5) × 10⁴ = 8 × 10⁴

Answer: 8 × 10⁴

Example 2: (6 × 10⁵) + (2 × 10³)

Solution: Different exponents! Convert to same power of 10:

Option 1: Use 10⁵ 2 × 10³ = 0.02 × 10⁵

Now add: (6 + 0.02) × 10⁵ = 6.02 × 10⁵

Answer: 6.02 × 10⁵

Example 3: (7 × 10⁴) - (3 × 10³)

Solution: Convert to 10⁴: 3 × 10³ = 0.3 × 10⁴

Subtract: (7 - 0.3) × 10⁴ = 6.7 × 10⁴

Answer: 6.7 × 10⁴

Example 4: (5.2 × 10⁶) + (8.5 × 10⁶)

Solution: Same exponents: (5.2 + 8.5) × 10⁶ = 13.7 × 10⁶

Adjust to proper form: 13.7 × 10⁶ = 1.37 × 10⁷

Answer: 1.37 × 10⁷

Combined Operations

Example 1: (4 × 10³) × (6 × 10²) ÷ (3 × 10⁴)

Solution: Step 1: Multiply first two numbers (4 × 6) × 10³⁺² = 24 × 10⁵ = 2.4 × 10⁶

Step 2: Divide by third number (2.4 × 10⁶) ÷ (3 × 10⁴) = (2.4 ÷ 3) × 10⁶⁻⁴ = 0.8 × 10² = 8 × 10¹

Answer: 8 × 10¹ or 80

Example 2: (5 × 10⁴) + (2 × 10⁴) - (3 × 10³)

Solution: Step 1: Add first two (same exponents) (5 + 2) × 10⁴ = 7 × 10⁴

Step 2: Subtract third (convert exponents) 3 × 10³ = 0.3 × 10⁴ 7 × 10⁴ - 0.3 × 10⁴ = 6.7 × 10⁴

Answer: 6.7 × 10⁴

Real-World Applications

Astronomy: Distance calculations

• Earth to Sun: 1.5 × 10⁸ km
• Sun to nearest star: 4 × 10¹³ km
• Ratio: (4 × 10¹³) ÷ (1.5 × 10⁸) ≈ 2.67 × 10⁵ times farther

Biology: Cell measurements

• Red blood cell: 7 × 10⁻⁶ m diameter
• Bacterial cell: 2 × 10⁻⁶ m diameter
• Difference: (7 - 2) × 10⁻⁶ = 5 × 10⁻⁶ m

Computing: Data calculations

• Internet traffic: 4.5 × 10¹⁸ bytes per day
• Average per second: (4.5 × 10¹⁸) ÷ (8.64 × 10⁴) ≈ 5.2 × 10¹³ bytes/sec

Chemistry: Molecular quantities

• Avogadro's number: 6.02 × 10²³ molecules/mole
• Two moles: 2 × (6.02 × 10²³) = 1.204 × 10²⁴ molecules

Calculator Tips

Most calculators have a scientific notation button (often labeled EE or EXP):

To enter 3.5 × 10⁸:

• Type: 3.5 [EE] 8
• Display might show: 3.5 E8 or 3.5⁺⁰⁸

To enter 2.7 × 10⁻⁵:

• Type: 2.7 [EE] [-] 5
• Display: 2.7 E-5 or 2.7⁻⁰⁵

Note: Don't type the ×10 part — the EE button does that!

Common Mistakes to Avoid

❌ Mistake 1: Adding exponents when adding numbers

• Wrong: (3 × 10⁴) + (2 × 10⁵) = 5 × 10⁹
• Right: Convert to same exponent first!

❌ Mistake 2: Forgetting to adjust after multiplication

• Wrong: (5 × 10³) × (4 × 10²) = 20 × 10⁵ (not proper form!)
• Right: 20 × 10⁵ = 2 × 10⁶

❌ Mistake 3: Subtracting exponents when multiplying

• Wrong: (2 × 10⁴) × (3 × 10⁵) = 6 × 10⁻¹
• Right: 6 × 10⁹ (add exponents!)

❌ Mistake 4: Not converting to same exponent before adding

• Wrong: (5 × 10⁶) + (3 × 10⁴) = 8 × 10⁶
• Right: 5.03 × 10⁶ or 503 × 10⁴

❌ Mistake 5: Forgetting negative sign in exponent

• Wrong: 10⁴ ÷ 10⁶ = 10²
• Right: 10⁴⁻⁶ = 10⁻²

Step-by-Step Strategy

For Multiplication:

1. Multiply the coefficients
2. Add the exponents
3. Adjust to proper form (1 ≤ a < 10)

For Division:

1. Divide the coefficients
2. Subtract the exponents
3. Adjust to proper form

For Addition/Subtraction:

1. Make exponents the same
2. Add or subtract coefficients
3. Keep the common exponent
4. Adjust to proper form

Quick Reference

Multiplication: (a × 10ᵐ) × (b × 10ⁿ) = (a × b) × 10ᵐ⁺ⁿ

Division: (a × 10ᵐ) ÷ (b × 10ⁿ) = (a ÷ b) × 10ᵐ⁻ⁿ

Addition/Subtraction: First make exponents equal, then: (a × 10ⁿ) ± (b × 10ⁿ) = (a ± b) × 10ⁿ

Proper Form: 1 ≤ coefficient < 10

Summary

Operations with scientific notation follow clear patterns:

Multiply: Multiply coefficients, add exponents

Divide: Divide coefficients, subtract exponents

Add/Subtract: Match exponents first, then add/subtract coefficients

Always adjust your answer to proper scientific notation (coefficient between 1 and 10).

These skills are essential for:

• Science calculations (astronomy, chemistry, biology)
• Engineering (very large and very small measurements)
• Technology (data sizes, processing speeds)
• Finance (large-scale economics)

Master these operations and you'll handle numbers of any size with confidence!