Describe How You Would Simplify The Given Expression.

Simplifying Algebraic Expressions: A Comprehensive Guide

Simplifying algebraic expressions is a fundamental skill in mathematics, crucial for solving equations, understanding functions, and tackling more complex problems. It involves manipulating the expression to make it more concise and easier to understand without changing its overall value. This guide provides a comprehensive walkthrough of simplification techniques, covering various scenarios with detailed examples. We'll explore the order of operations, combining like terms, factoring, and expanding expressions, equipping you with the tools to confidently simplify any algebraic expression.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before diving into simplification, it's crucial to understand the order of operations, often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). These acronyms dictate the sequence in which operations should be performed:

1. Parentheses/Brackets: Evaluate any expressions within parentheses or brackets first. Work from the innermost set outwards.

2. Exponents/Orders: Calculate any exponents or powers.

3. Multiplication and Division: Perform multiplication and division from left to right. These operations have equal precedence.

4. Addition and Subtraction: Perform addition and subtraction from left to right. These operations also have equal precedence.

Example:

Simplify: 3 + 2 × (4 - 1)² + 5 ÷ 5

1. Parentheses: 4 - 1 = 3
2. Exponents: 3² = 9
3. Multiplication: 2 × 9 = 18
4. Division: 5 ÷ 5 = 1
5. Addition: 3 + 18 + 1 = 22

Therefore, the simplified expression is 22.

Combining Like Terms

Like terms are terms that have the same variables raised to the same powers. To combine like terms, simply add or subtract their coefficients (the numerical part of the term).

Example:

Simplify: 3x² + 5x - 2x² + 7x + 4

1. Identify like terms: 3x² and -2x² are like terms; 5x and 7x are like terms.
2. Combine like terms: (3x² - 2x²) + (5x + 7x) + 4 = x² + 12x + 4

The simplified expression is x² + 12x + 4.

Distributive Property

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

Example:

Simplify: 2(3x + 4) - 5(x - 2)

1. Distribute: 2(3x) + 2(4) - 5(x) - 5(-2) = 6x + 8 - 5x + 10
2. Combine like terms: (6x - 5x) + (8 + 10) = x + 18

The simplified expression is x + 18.

Factoring

Factoring is the reverse of the distributive property. It involves expressing an expression as a product of simpler factors. Common factoring techniques include:

• Greatest Common Factor (GCF): Find the largest factor that divides all terms in the expression.

Example:

Simplify: 6x² + 9x

1. Find the GCF: The GCF of 6x² and 9x is 3x.
2. Factor out the GCF: 3x(2x + 3)

The factored expression is 3x(2x + 3).

• Difference of Squares: a² - b² = (a + b)(a - b)

Example:

Simplify: x² - 16

1. Recognize the difference of squares: x² is a² and 16 is b² (where a = x and b = 4).
2. Factor: (x + 4)(x - 4)

The factored expression is (x + 4)(x - 4).

• Trinomial Factoring: Factoring quadratic trinomials (ax² + bx + c) often involves finding two numbers that add up to b and multiply to ac.

Example:

Simplify: x² + 5x + 6

1. Find two numbers: The numbers 2 and 3 add up to 5 and multiply to 6.
2. Factor: (x + 2)(x + 3)

The factored expression is (x + 2)(x + 3).

Expanding Expressions with Multiple Variables and Exponents

Simplifying expressions with multiple variables and exponents involves applying the same principles as before, but with extra attention to the rules of exponents. Remember:

• xᵐ × xⁿ = xᵐ⁺ⁿ (When multiplying terms with the same base, add the exponents)
• xᵐ ÷ xⁿ = xᵐ⁻ⁿ (When dividing terms with the same base, subtract the exponents)
• (xᵐ)ⁿ = xᵐⁿ (When raising a power to a power, multiply the exponents)

Example:

Simplify: 2x²(3xy³ - 4x²y) + 5x³y³

1. Distribute: 6x³y³ - 8x⁴y + 5x³y³
2. Combine like terms: (6x³y³ + 5x³y³) - 8x⁴y = 11x³y³ - 8x⁴y

The simplified expression is 11x³y³ - 8x⁴y.

Simplifying Expressions with Fractions

When dealing with fractions, remember to find a common denominator when adding or subtracting fractions. Also, simplify fractions by canceling common factors in the numerator and denominator.

Example:

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

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

The simplified expression is (5/6)x.

Simplifying Expressions with Radicals

Simplifying expressions with radicals involves simplifying the radicand (the expression inside the radical) and then simplifying the radical itself. Remember to look for perfect square factors within the radicand.

Example:

Simplify: √(12x⁴y²)

1. Find perfect square factors: 12 = 4 × 3; x⁴ = (x²)²; y² = y²
2. Simplify: √(4 × 3 × (x²)² × y²) = √4 × √(x²)² × √y² × √3 = 2x²y√3

The simplified expression is 2x²y√3.

Handling Absolute Value Expressions

Absolute value expressions represent the distance of a number from zero. Therefore, the result is always non-negative. When simplifying, consider the possible cases based on the sign of the expression inside the absolute value.

Example:

Simplify: |x - 3| if x > 3

If x > 3, then (x - 3) is positive. Therefore, |x - 3| = x - 3.

Advanced Simplification Techniques

For more complex expressions, you may need to apply multiple techniques in combination. This might involve factoring, expanding, combining like terms, and simplifying fractions or radicals sequentially. Always prioritize the order of operations and systematically work through each step.

Conclusion

Simplifying algebraic expressions is a crucial skill in mathematics. By mastering the techniques outlined above—the order of operations, combining like terms, distributing, factoring, and handling various expression types—you'll be able to confidently tackle even the most complex expressions and progress to more advanced mathematical concepts. Remember to practice regularly and break down complex problems into smaller, manageable steps. This consistent practice will build your confidence and proficiency in simplifying algebraic expressions.