Describe How You Would Simplify The Given Expression. Es002-1.jpg

I cannot access local files, including the image file "es002-1.jpg." Therefore, I cannot provide a specific simplification of the expression shown in that image. However, I can give you a comprehensive guide on how to simplify various types of mathematical expressions, covering many common scenarios you might encounter. This will equip you with the knowledge to tackle most simplification problems, including the one in your image, once you provide the expression itself.

Mastering Mathematical Expression Simplification: A Comprehensive Guide

Simplifying mathematical expressions is a fundamental skill in algebra and beyond. It involves using various techniques to rewrite a complex expression in a simpler, more manageable form without changing its value. This guide will cover various strategies, along with examples, to help you master this essential skill.

1. Understanding the Order of Operations (PEMDAS/BODMAS)

Before attempting any simplification, you must understand the order of operations. This ensures that you perform calculations in the correct sequence, avoiding errors. The acronyms PEMDAS and BODMAS represent the same order:

• Parentheses/Brackets: Perform operations within parentheses or brackets first.
• Exponents/Orders: Evaluate exponents (powers) next.
• Multiplication and Division: Perform multiplication and division from left to right.
• Addition and Subtraction: Perform addition and subtraction from left to right.

Example:

Simplify: 3 + 2 × (4 - 1)²

1. Parentheses: 4 - 1 = 3
2. Exponents: 3² = 9
3. Multiplication: 2 × 9 = 18
4. Addition: 3 + 18 = 21

Therefore, the simplified expression is 21.

2. Combining Like Terms

Like terms are terms that have the same variables raised to the same powers. You can combine like terms by adding or subtracting their coefficients (the numbers in front of the variables).

Example:

Simplify: 3x² + 5x - 2x² + 7x + 1

1. Identify like terms: 3x² and -2x² are like terms; 5x and 7x are like terms.
2. Combine like terms: (3x² - 2x²) + (5x + 7x) + 1 = x² + 12x + 1

The simplified expression is x² + 12x + 1.

3. Distributive Property

The distributive property states that a(b + c) = ab + ac. This allows you to expand expressions by multiplying a term outside parentheses by each term inside.

Example:

Simplify: 2(x + 3y - 4)

1. Apply the distributive property: 2(x) + 2(3y) + 2(-4)
2. Simplify: 2x + 6y - 8

The simplified expression is 2x + 6y - 8.

4. Factoring

Factoring is the reverse of the distributive property. It involves finding common factors among terms and rewriting the expression as a product of factors. This is crucial for simplifying fractions and solving equations.

Example:

Simplify: 4x² + 8x

1. Find the greatest common factor (GCF): The GCF of 4x² and 8x is 4x.
2. Factor out the GCF: 4x(x + 2)

The simplified expression is 4x(x + 2).

5. Working with Fractions

Simplifying expressions involving fractions requires using rules for fraction arithmetic:

• Adding/Subtracting Fractions: Find a common denominator.
• Multiplying Fractions: Multiply numerators and denominators separately.
• Dividing Fractions: Invert the second fraction and multiply.

Example:

Simplify: (2/3)x + (1/6)x

1. Find a common denominator: The common denominator of 3 and 6 is 6.
2. Rewrite fractions with the common denominator: (4/6)x + (1/6)x
3. Add the fractions: (5/6)x

The simplified expression is (5/6)x.

6. Simplifying Radicals

Radicals (square roots, cube roots, etc.) can often be simplified by factoring the radicand (the number under the radical symbol) and applying the property √(ab) = √a × √b.

Example:

Simplify: √12

1. Factor the radicand: 12 = 4 × 3
2. Apply the property: √12 = √(4 × 3) = √4 × √3 = 2√3

The simplified expression is 2√3.

7. Exponent Rules

Understanding exponent rules is crucial for simplifying expressions with exponents:

• Product Rule: aᵐ × aⁿ = aᵐ⁺ⁿ
• Quotient Rule: aᵐ / aⁿ = aᵐ⁻ⁿ
• Power Rule: (aᵐ)ⁿ = aᵐⁿ
• Zero Exponent: a⁰ = 1 (a ≠ 0)
• Negative Exponent: a⁻ⁿ = 1/aⁿ

Example:

Simplify: x³ × x⁻²

1. Apply the product rule: x³⁺⁽⁻²⁾ = x¹ = x

The simplified expression is x.

8. Combining Techniques

Often, simplifying an expression requires combining several of the techniques described above. Always follow the order of operations and strategically apply the relevant rules.

Example:

Simplify: 3(x + 2)² - 4x + 6

1. Parentheses (exponent first): (x + 2)² = x² + 4x + 4
2. Distributive Property: 3(x² + 4x + 4) = 3x² + 12x + 12
3. Combine Like Terms: 3x² + 12x + 12 - 4x + 6 = 3x² + 8x + 18

The simplified expression is 3x² + 8x + 18.

Conclusion: A Step-by-Step Approach to Simplification

To effectively simplify any mathematical expression, follow these steps:

1. Analyze the Expression: Identify the types of terms and operations involved.
2. Apply the Order of Operations: Always follow PEMDAS/BODMAS.
3. Use Relevant Simplification Techniques: Apply the distributive property, factoring, combining like terms, simplifying fractions and radicals, and exponent rules as needed.
4. Check your work: Ensure your simplification is correct by substituting values for the variables and comparing the results of both the original and simplified expressions.

Remember, practice is key to mastering expression simplification. Work through numerous examples, gradually increasing the complexity of the expressions you tackle. By understanding these techniques and practicing consistently, you’ll develop the skills to simplify even the most challenging mathematical expressions. Please provide the expression from your image "es002-1.jpg" and I will be happy to help you simplify it specifically.