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

I do not have access to local files, including images, so I cannot describe how to simplify the expression shown in "es002-1.jpg". To help you simplify a mathematical expression, please provide the expression itself in the text of your prompt. I will then provide a step-by-step explanation of how to simplify it, along with explanations of the underlying mathematical principles.

However, I can offer a general guide on simplifying mathematical expressions and cover various techniques and examples. This will help you tackle a wide range of simplification problems, regardless of the specific expression in your image.

Simplifying Mathematical Expressions: A Comprehensive Guide

Simplifying mathematical expressions is a fundamental skill in algebra and other branches of mathematics. It involves manipulating an expression to make it less complex while maintaining its original value. This often makes the expression easier to understand, evaluate, and use in further calculations. The process involves applying various algebraic rules and techniques.

Key Principles and Techniques

Several core principles guide the simplification process:

• Order of Operations (PEMDAS/BODMAS): This dictates the sequence of operations: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Consistent application is crucial for accurate simplification.

• Combining Like Terms: Terms with the same variables raised to the same powers can be combined by adding or subtracting their coefficients. For example, 3x + 2x = 5x and 7y² - 4y² = 3y².

• Distributive Property: This allows you to expand expressions by multiplying a term outside parentheses by each term inside the parentheses. For example, a(b + c) = ab + ac.

• Factoring: This is the reverse of the distributive property. It involves finding common factors among terms and expressing the expression as a product of these factors. For example, ab + ac = a(b + c).

• Cancelling Common Factors: In fractions, if the numerator and denominator share a common factor, you can cancel them out. For example, (6x²y) / (3xy) = 2x (assuming x and y are not zero).

• Exponent Rules: These rules govern how to simplify expressions with exponents. They include rules for multiplying exponents (a^m * aⁿ = a^m+n), dividing exponents (a^m / aⁿ = a^m-n), raising a power to a power ((a^m)ⁿ = a^mn), and others.

Step-by-Step Approach to Simplifying Expressions

Let's illustrate these principles with examples:

Example 1: Simplifying a Linear Expression

Simplify: 3(x + 2) - 2x + 5

1. Distributive Property: 3(x + 2) = 3x + 6

2. Substitute: The expression becomes 3x + 6 - 2x + 5

3. Combine Like Terms: (3x - 2x) + (6 + 5) = x + 11

Therefore, the simplified expression is x + 11.

Example 2: Simplifying a Quadratic Expression

Simplify: 2x² + 4x - x² + 3x - 7

1. Combine Like Terms: (2x² - x²) + (4x + 3x) - 7 = x² + 7x - 7

Therefore, the simplified expression is x² + 7x - 7.

Example 3: Simplifying a Rational Expression

Simplify: (6x²y³z) / (2xyz)

1. Cancel Common Factors: We can cancel out 2, x, y, and z from both the numerator and the denominator.

2. Simplify: (6x²y³z) / (2xyz) = 3xy²

Therefore, the simplified expression is 3xy².

Example 4: Simplifying an Expression with Exponents

Simplify: (2x³y²)² * (xy)³

1. Power of a Product Rule: (2x³y²)² = 4x⁶y⁴ and (xy)³ = x³y³

2. Multiplying Exponents: 4x⁶y⁴ * x³y³ = 4x⁶⁺³y⁴⁺³ = 4x⁹y⁷

Therefore, the simplified expression is 4x⁹y⁷.

Example 5: Simplifying a more complex expression

Simplify: [(4x² - 16) / (2x + 4)] + (x - 2)

1. Factor the numerator: 4x² - 16 = 4(x² - 4) = 4(x - 2)(x + 2)

2. Factor the denominator: 2x + 4 = 2(x + 2)

3. Simplify the fraction: [4(x - 2)(x + 2)] / [2(x + 2)] = 2(x - 2)

4. Add the remaining term: 2(x - 2) + (x - 2) = 2x - 4 + x - 2 = 3x - 6

Therefore, the simplified expression is 3x - 6.

Dealing with Different Types of Expressions

The techniques described above are applicable to various types of expressions, including:

• Polynomial Expressions: These involve sums of terms with variables raised to non-negative integer powers. Simplifying often involves combining like terms and applying the distributive property.

• Rational Expressions: These are expressions in the form of fractions where both the numerator and denominator are polynomial expressions. Simplification involves factoring and canceling common factors.

• Radical Expressions: These involve square roots, cube roots, and other roots. Simplification often involves simplifying the radicand and applying rules for simplifying radicals.

• Exponential Expressions: These involve exponents. Simplifying uses the rules of exponents.

• Trigonometric Expressions: These involve trigonometric functions. Simplifying often uses trigonometric identities.

Remember that the key to successfully simplifying expressions is to work systematically, applying the correct order of operations and using the appropriate algebraic techniques. Practice is crucial to mastering these skills. Once you provide the expression from your image, I can provide a much more targeted and specific guide.