Operations with Scientific Notation

Scientific notation is incredibly useful for working with very large or very small numbers! Once you know how to add, subtract, multiply, and divide numbers in scientific notation, you can tackle problems in science, engineering, and technology with ease.

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Review: What Is Scientific Notation?

Scientific notation expresses numbers as:

a × 10ⁿ

Where:

• a is a number between 1 and 10 (1 ≤ a < 10)
• n is an integer exponent

Examples:

• 3,000 = 3 × 10³
• 0.0045 = 4.5 × 10⁻³
• 6,500,000 = 6.5 × 10⁶

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Multiplying Scientific Notation

Rule: Multiply the coefficients, add the exponents

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

Example 1: (3 × 10⁴) × (2 × 10⁵)

Solution: Step 1: Multiply coefficients: 3 × 2 = 6 Step 2: Add exponents: 10⁴⁺⁵ = 10⁹ Step 3: Combine: 6 × 10⁹

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

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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

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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

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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

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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

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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!

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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

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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

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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

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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!