Write The Expression In Simplest Form.

Write the Expression in Simplest Form: A Comprehensive Guide

Simplifying expressions is a fundamental skill in mathematics, crucial for solving equations, understanding relationships between variables, and generally making mathematical work easier to manage. This guide will delve into various techniques for simplifying expressions, covering everything from basic arithmetic operations to more complex algebraic manipulations. We’ll explore the order of operations, combining like terms, factoring, and working with fractions and exponents. By the end, you’ll have a robust understanding of how to write expressions in their simplest form.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we begin simplifying, 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). Both acronyms represent the same order:

• Parentheses/Brackets: Always perform operations within parentheses or brackets first. Work from the innermost parentheses outwards.
• Exponents/Orders: Next, evaluate any exponents or powers.
• Multiplication and Division: Perform multiplication and division from left to right. These operations have equal precedence.
• Addition and Subtraction: Finally, perform addition and subtraction from left to right. These operations also have equal precedence.

Example:

Simplify the expression: 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 numbers in front of the variables).

Example:

Simplify the expression: 5x + 3y - 2x + 7y

1. Identify like terms: 5x and -2x are like terms; 3y and 7y are like terms.
2. Combine like terms: 5x - 2x = 3x and 3y + 7y = 10y
3. Write the simplified expression: 3x + 10y

Factoring Expressions

Factoring involves expressing an expression as a product of simpler expressions. This is a powerful technique for simplifying expressions and solving equations. Common factoring techniques include:

• Greatest Common Factor (GCF): Find the largest factor common to all terms and factor it out.

Example:

Simplify the expression: 6x² + 12x

1. Find the GCF: The GCF of 6x² and 12x is 6x.
2. Factor out the GCF: 6x(x + 2)

• Difference of Squares: This applies to expressions of the form a² - b², which can be factored as (a + b)(a - b).

Example:

Simplify the expression: x² - 9

1. Recognize the difference of squares: x² - 9 = x² - 3²
2. Factor: (x + 3)(x - 3)

• Trinomial Factoring: Factoring trinomials (expressions with three terms) often involves finding two numbers that add up to the coefficient of the middle term and multiply to the product of the coefficient of the first and last terms. This can be quite involved and requires practice.

Example:

Simplify the expression: 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)

Working with Fractions

Simplifying expressions involving fractions requires a solid understanding of fraction arithmetic. Key principles include:

• Simplifying fractions: Reduce fractions to their lowest terms by dividing both the numerator and denominator by their greatest common factor.

Example:

Simplify the fraction: 12/18

1. Find the GCF: The GCF of 12 and 18 is 6.
2. Simplify: 12/18 = (12 ÷ 6) / (18 ÷ 6) = 2/3

• Adding and subtracting fractions: Find a common denominator before adding or subtracting fractions.

Example:

Simplify the expression: 1/2 + 1/3

1. Find a common denominator: The least common multiple of 2 and 3 is 6.
2. Rewrite the fractions: 1/2 = 3/6 and 1/3 = 2/6
3. Add: 3/6 + 2/6 = 5/6

• Multiplying fractions: Multiply the numerators together and the denominators together.

Example:

Simplify the expression: (2/3) × (3/4)

1. Multiply: (2 × 3) / (3 × 4) = 6/12
2. Simplify: 6/12 = 1/2

• Dividing fractions: Invert the second fraction and multiply.

Example:

Simplify the expression: (2/3) ÷ (1/2)

1. Invert and multiply: (2/3) × (2/1) = 4/3

Working with Exponents

Exponents represent repeated multiplication. Understanding exponent rules is essential for simplifying expressions with exponents:

• Product Rule: aᵐ × aⁿ = aᵐ⁺ⁿ (when multiplying terms with the same base, add the exponents)

Example:

Simplify the expression: x³ × x²

1. Apply the product rule: x³⁺² = x⁵

• Quotient Rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (when dividing terms with the same base, subtract the exponents)

Example:

Simplify the expression: x⁵ ÷ x²

1. Apply the quotient rule: x⁵⁻² = x³

• Power Rule: (aᵐ)ⁿ = aᵐⁿ (when raising a power to a power, multiply the exponents)

Example:

Simplify the expression: (x²)³

1. Apply the power rule: x²ˣ³ = x⁶

• Zero Exponent: a⁰ = 1 (any non-zero base raised to the power of zero is 1)

Example:

Simplify the expression: x⁰

1. Apply the zero exponent rule: 1

Simplifying Expressions with Multiple Operations

Many expressions require the application of multiple simplification techniques. It's crucial to follow the order of operations and apply the appropriate techniques in a systematic manner.

Example:

Simplify the expression: 2(3x² + 4x - 6) - 3(x² - 2x + 5)

1. Distribute: 6x² + 8x - 12 - 3x² + 6x - 15
2. Combine like terms: (6x² - 3x²) + (8x + 6x) + (-12 - 15)
3. Simplify: 3x² + 14x - 27

Conclusion

Simplifying expressions is a core skill in algebra and beyond. By mastering the techniques outlined in this guide – understanding the order of operations, combining like terms, factoring, working with fractions and exponents – you'll significantly improve your ability to manipulate mathematical expressions, solve equations, and grasp more complex mathematical concepts. Remember, practice is key. The more you work through different types of expressions, the more confident and efficient you'll become in simplifying them to their simplest form. This detailed guide provides a strong foundation; continue practicing and exploring more advanced techniques to further enhance your mathematical abilities.