What Is The Simplified Form Of The Following Expression

Simplifying Mathematical Expressions: A Comprehensive Guide

Simplifying mathematical expressions is a fundamental skill in algebra and beyond. It involves rewriting a complex expression in a more concise and manageable form without changing its value. This process is crucial for solving equations, understanding relationships between variables, and making calculations easier. This article will explore various techniques for simplifying expressions, focusing on common algebraic manipulations, and provide detailed examples to solidify your understanding. We'll cover everything from basic arithmetic operations to more advanced techniques like factoring and using the order of operations (PEMDAS/BODMAS).

I. Understanding the Fundamentals: Order of Operations (PEMDAS/BODMAS)

Before diving into complex simplification, it's crucial to understand the order of operations. This ensures consistency in evaluating expressions and arriving at the correct simplified form. The acronyms PEMDAS and BODMAS are commonly used to remember the order:

• PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
• BODMAS: Brackets, Orders (exponents), Division and Multiplication (from left to right), Addition and Subtraction (from left to right).

Both acronyms represent the same order of operations; the only difference lies in the terminology used. Let's illustrate this with an example:

Example: Simplify 2 + 3 × 4 – (5 + 2)² ÷ 7

1. Parentheses/Brackets: First, we simplify the expression inside the parentheses: (5 + 2) = 7. The expression becomes: 2 + 3 × 4 – 7² ÷ 7.
2. Exponents/Orders: Next, we calculate the exponent: 7² = 49. The expression becomes: 2 + 3 × 4 – 49 ÷ 7.
3. Multiplication and Division (from left to right): Now, we perform multiplication and division in the order they appear: 3 × 4 = 12 and 49 ÷ 7 = 7. The expression becomes: 2 + 12 – 7.
4. Addition and Subtraction (from left to right): Finally, we perform addition and subtraction: 2 + 12 = 14, and 14 – 7 = 7.

Therefore, the simplified form of the expression 2 + 3 × 4 – (5 + 2)² ÷ 7 is 7.

II. Simplifying Expressions with Variables

Simplifying expressions involving variables requires combining like terms and applying distributive properties. Like terms are terms that have the same variables raised to the same powers. For example, 3x and 5x are like terms, but 3x and 3x² are not.

A. Combining Like Terms:

Example: Simplify 5x + 2y – 3x + 4y

1. Identify like terms: 5x and –3x are like terms, and 2y and 4y are like terms.
2. Combine like terms: (5x – 3x) + (2y + 4y) = 2x + 6y.

The simplified expression is 2x + 6y.

B. Distributive Property:

The distributive property states that a(b + c) = ab + ac. This property is crucial for removing parentheses and simplifying expressions.

Example: Simplify 3(x + 2) – 2(x – 1)

1. Distribute the 3 and the –2: 3(x) + 3(2) – 2(x) – 2(–1) = 3x + 6 – 2x + 2.
2. Combine like terms: (3x – 2x) + (6 + 2) = x + 8.

The simplified expression is x + 8.

III. Factoring Expressions

Factoring is the reverse of the distributive property. It involves expressing an expression as a product of simpler expressions. Factoring is essential for solving equations and simplifying more complex expressions. Common factoring techniques include:

• Greatest Common Factor (GCF): Finding the largest common factor among the terms and factoring it out.
• Difference of Squares: Factoring expressions of the form a² – b² as (a + b)(a – b).
• Trinomial Factoring: Factoring quadratic expressions of the form ax² + bx + c.

A. Greatest Common Factor (GCF):

Example: Factor 6x² + 9x

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

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

B. Difference of Squares:

Example: Factor x² – 25

1. Recognize that this is a difference of squares (x² – 5²).
2. Apply the formula: (x + 5)(x – 5).

The factored expression is (x + 5)(x – 5).

C. Trinomial Factoring:

Example: Factor x² + 5x + 6

We need to find two numbers that add up to 5 (the coefficient of x) and multiply to 6 (the constant term). These numbers are 2 and 3.

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

IV. Simplifying Rational Expressions

Rational expressions are expressions that can be written as a fraction where the numerator and denominator are polynomials. Simplifying rational expressions involves canceling common factors in the numerator and denominator.

Example: Simplify (x² – 4) / (x + 2)

1. Factor the numerator: x² – 4 = (x + 2)(x – 2).
2. Rewrite the expression: [(x + 2)(x – 2)] / (x + 2).
3. Cancel the common factor (x + 2): x – 2.

The simplified expression is x – 2, provided x ≠ -2 (to avoid division by zero).

V. Simplifying Expressions with Radicals (Roots)

Simplifying expressions with radicals involves simplifying the radicand (the number inside the radical) and removing perfect square factors from under the radical.

Example: Simplify √75

1. Find the prime factorization of 75: 75 = 3 × 5 × 5 = 3 × 5².
2. Rewrite the expression: √(3 × 5²) = √3 × √5² = 5√3.

The simplified expression is 5√3.

VI. Advanced Techniques and Considerations:

Simplifying expressions can involve more complex techniques, depending on the nature of the expression. These may include:

• Partial Fraction Decomposition: Breaking down complex rational expressions into simpler fractions.
• Complex Number Arithmetic: Simplifying expressions involving imaginary units (i).
• Logarithmic and Exponential Simplifications: Using logarithmic and exponential properties to simplify expressions.

Mastering these techniques requires a solid understanding of algebraic concepts and practice.

VII. Frequently Asked Questions (FAQ)

• Q: What is the difference between simplifying and solving?

  • A: Simplifying an expression means rewriting it in a more concise form without changing its value. Solving an equation means finding the value of the variable that makes the equation true.

• Q: Can I simplify an expression in multiple ways?

  • A: Often, there are multiple ways to simplify an expression, but the final simplified form should be equivalent. Different approaches might involve different intermediate steps, but the final answer should be the same.

• Q: What if I make a mistake during simplification?

  • A: Carefully review your steps. Check your arithmetic, ensure you have applied the order of operations correctly, and double-check your factoring or other manipulations. Practice is key to minimizing errors.

• Q: How can I improve my skills in simplifying expressions?

  • A: The best way to improve is through consistent practice. Work through numerous examples, try different approaches, and seek help when needed. Understanding the underlying principles is crucial for mastering the techniques.

VIII. Conclusion

Simplifying mathematical expressions is a crucial skill for success in mathematics and related fields. By understanding the order of operations, combining like terms, applying distributive and factoring techniques, and mastering the simplification of rational expressions and radicals, you can significantly enhance your ability to solve problems and understand mathematical relationships. Consistent practice and a focus on understanding the underlying principles are key to mastering this fundamental skill. Remember, the goal is not just to get the right answer, but to develop a deep understanding of the underlying processes involved. With dedicated effort and practice, simplifying expressions will become second nature.