Lesson 71: Multiplying and Dividing Scientific Notation (Combined Operations)

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• Lesson Description
• Learning Objectives
• Topics Covered
• Key Terms
• Lesson Examples
• Practice Problems
• Tips & Common Mistakes
• Summary
• Challenge Problems
• Unit Navigation

Lesson Description

In this lesson, you will evaluate expressions that combine multiplication and division in scientific notation. You will apply the product rule and quotient rule of exponents, simplify coefficients carefully, and ensure the final answer is written in proper scientific notation form.

Learning Objectives

• Apply multiplication and division rules for powers of ten in a single expression.
• Simplify coefficients accurately before adjusting exponents.
• Rewrite final answers in proper scientific notation.

Topics Covered

• Product Rule of Exponents
• Quotient Rule of Exponents
• Order of operations with scientific notation

Key Terms

• Product Rule: When multiplying powers with the same base, add the exponents.
• Quotient Rule: When dividing powers with the same base, subtract the exponents.
• Proper Scientific Notation: A number written as a coefficient between 1 and 10 multiplied by a power of 10.

Lesson Examples

Example 1: Multiply Then Divide

1. Simplify: (2 × 10³)(3 × 10⁴) ÷ (6 × 10²).
2. Multiply coefficients: 2 × 3 = 6. Add exponents: 10³ × 10⁴ = 10⁷.
3. Divide by (6 × 10²): 6 ÷ 6 = 1 and subtract exponents: 10⁷ ÷ 10² = 10⁵. Final answer: 1 × 10⁵.

Example 2: Mixed Negative Exponents

1. Simplify: (4 × 10⁻²)(5 × 10³) ÷ (2 × 10¹).
2. Multiply: 4 × 5 = 20 and add exponents: 10⁻² × 10³ = 10¹.
3. Divide: 20 ÷ 2 = 10 and subtract exponents: 10¹ ÷ 10¹ = 10⁰. Since 10 × 10⁰ = 10, rewrite as 1 × 10¹.

Practice Problems

1. (3 × 10²)(4 × 10⁵) ÷ (6 × 10³)
2. (5 × 10⁻¹)(2 × 10⁴) ÷ (5 × 10²)
3. (7 × 10³)(8 × 10⁻²) ÷ (4 × 10¹)
4. (9 × 10⁶)(2 × 10⁻³) ÷ (3 × 10²)

Tips & Common Mistakes

• Follow order of operations carefully.
• Add exponents when multiplying, subtract when dividing.
• Always adjust the coefficient to stay between 1 and 10.

Summary

When multiplying and dividing scientific notation in the same expression, multiply coefficients and add exponents first, then divide coefficients and subtract exponents as needed. Always confirm the final answer is written in proper scientific notation.

Challenge Problems

1. (1.5 × 10⁴)(2 × 10³) ÷ (3 × 10²)
2. (6 × 10⁻³)(4 × 10⁵) ÷ (8 × 10¹)
3. (2.5 × 10⁷)(4 × 10⁻²) ÷ (5 × 10³)