Lesson 92: Nested Parentheses

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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 teaches students how to approach and solve mathematical expressions that include nested parentheses. We will carefully apply PEMDAS, solving the innermost parentheses first, then moving outward to ensure accurate results.

Learning Objectives

• Identify nested parentheses within a mathematical expression.
• Apply the correct order of operations to expressions with multiple layers of parentheses.
• Solve multi-step problems accurately by working from the innermost parentheses outward.

Topics Covered

• PEMDAS applied to nested parentheses
• Stepwise simplification of multi-layered expressions
• Combining parentheses with fractions, decimals, and negative numbers

Key Terms

• Nested Parentheses: Parentheses that are contained within other parentheses.
• Innermost Parentheses: The parentheses that are solved first in a nested expression.
• Order of Operations (PEMDAS): The set of rules dictating the sequence of operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

Lesson Examples

Example 1: Simple Nested Parentheses

1. Problem: 2 × (3 + (4 − 1))
2. Step 1: Solve innermost parentheses: (4 − 1) = 3
3. Step 2: Replace and simplify outer parentheses: 3 + 3 = 6
4. Step 3: Multiply by 2: 2 × 6 = 12

Example 2: Nested Parentheses with Decimals

1. Problem: 1.5 × (2 + (0.5 + 1.2))
2. Step 1: Solve innermost parentheses: (0.5 + 1.2) = 1.7
3. Step 2: Simplify outer parentheses: 2 + 1.7 = 3.7
4. Step 3: Multiply by 1.5: 1.5 × 3.7 = 5.55

Example 3: Multiple Nested Layers

1. Problem: ((2 + 3) × (4 − 1)) + 5
2. Step 1: Solve innermost parentheses: (2 + 3) = 5 and (4 − 1) = 3
3. Step 2: Multiply results: 5 × 3 = 15
4. Step 3: Add 5: 15 + 5 = 20

Practice Problems

1. 3 × (2 + (5 − 3))
2. (1.2 + (0.8 × 2)) − 1
3. ((4 − 2) + 3) × 2
4. 5 + (2 × (3 + (1 + 2)))

Tips & Common Mistakes

• Always start with the innermost parentheses before moving outward.
• Do not forget to follow PEMDAS for operations inside each layer of parentheses.
• Check your work after each step to avoid compounding errors in nested layers.

Summary

Nested parentheses require careful attention to the order of operations. Solving from the innermost parentheses outward ensures correct results, even when decimals, fractions, or negative numbers are involved. Following PEMDAS systematically avoids mistakes in multi-step expressions.

Challenge Problems

1. (2 + (3 × (4 − 2))) ÷ 2
2. 1.5 × ((2 + 3) − (0.5 × 2))
3. ((3 + 1) × (2 + (1 + 1))) − 4