Lesson 69: Exponents & Radicals Word Problems

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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

This lesson integrates exponent rules and radical concepts into practical word problems. Students will translate verbal descriptions into algebraic expressions and equations, then simplify or evaluate using exponent properties and square roots.

Learning Objectives

• Translate real-world situations into expressions involving exponents or radicals.
• Apply exponent rules to solve growth and area problems.
• Use square roots to solve geometric and measurement problems.

Topics Covered

• Area and volume problems using exponents
• Pythagorean Theorem applications
• Growth patterns and squared relationships

Key Terms

• Exponent: A number indicating how many times a base is multiplied by itself.
• Radical: A symbol (√) used to represent roots, such as square roots.
• Pythagorean Theorem: a² + b² = c², used to find the missing side of a right triangle.

Lesson Examples

Example 1: Area of a Square

1. A square has side length 7 cm. Area = s².
2. Substitute: 7².
3. Answer: 49 cm².

Example 2: Finding a Missing Side

1. A right triangle has legs 6 and 8. Use a² + b² = c².
2. 6² + 8² = 36 + 64 = 100.
3. c = √100 = 10.

Example 3: Population Growth

1. A population doubles each year. Starting at 5, after 3 years: 5 × 2³.
2. 2³ = 8.
3. Total population = 40.

Practice Problems

1. Find the area of a square with side length 9.
2. A right triangle has legs 5 and 12. Find the hypotenuse.
3. Evaluate 3 × 4².
4. If a number is squared and equals 81, what is the number?

Tips & Common Mistakes

• Carefully translate words into mathematical expressions.
• Square numbers before multiplying when required.
• Take square roots only after simplifying inside the radical.

Summary

Word problems involving exponents and radicals require careful translation from language to algebra. Apply exponent rules for repeated multiplication, and use square roots to reverse squaring operations. Always simplify fully and check units when solving real-world problems.

Challenge Problems

1. A square garden has area 144 m². Find the side length.
2. A right triangle has hypotenuse 13 and one leg 5. Find the other leg.
3. A bacteria culture triples every hour. Starting with 2 bacteria, how many after 4 hours?