Simplify The Expression Given Below.

Simplifying Algebraic Expressions: A Comprehensive Guide

Simplifying algebraic expressions is a fundamental skill in mathematics, crucial for solving equations, understanding relationships between variables, and tackling more complex problems. This comprehensive guide will walk you through the process, covering various techniques and providing plenty of examples to solidify your understanding. We'll break down the process step-by-step, making it accessible even to those new to algebra. By the end, you'll be confident in simplifying even the most challenging expressions.

Introduction: What are Algebraic Expressions?

An algebraic expression is a mathematical phrase that combines numbers, variables, and operators (like +, -, ×, ÷). Variables are letters that represent unknown numbers. For example, 2x + 3y - 5 is an algebraic expression. Simplifying an expression means rewriting it in a more concise and manageable form without changing its value. This involves combining like terms, using the distributive property, and applying the order of operations (PEMDAS/BODMAS).

The Fundamentals: Understanding Like Terms and the Distributive Property

Before we dive into simplifying complex expressions, let's review two key concepts:

• Like Terms: These are terms that have the same variables raised to the same powers. For instance, in the expression 3x + 2x - 5y + 7, 3x and 2x are like terms because they both have the variable x raised to the power of 1. Similarly, constants (numbers without variables) are also like terms.

• Distributive Property: This property states that a(b + c) = ab + ac. It allows us to multiply a term by a sum or difference within parentheses. The reverse process, factoring, is also essential for simplification. For example, 2(x + 3) can be expanded to 2x + 6, and conversely, 2x + 6 can be factored to 2(x + 3).

Step-by-Step Guide to Simplifying Algebraic Expressions

Let's illustrate the simplification process with examples of increasing complexity:

Example 1: Combining Like Terms

Simplify the expression: 4x + 7y - 2x + 3y

Steps:

1. Identify like terms: 4x and -2x are like terms; 7y and 3y are like terms.
2. Combine like terms: 4x - 2x = 2x and 7y + 3y = 10y
3. Write the simplified expression: 2x + 10y

Example 2: Using the Distributive Property

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

Steps:

1. Apply the distributive property: 3(2x + 5) = 6x + 15
2. Rewrite the expression: 6x + 15 - 4x
3. Combine like terms: 6x - 4x = 2x
4. Write the simplified expression: 2x + 15

Example 3: A More Complex Expression

Simplify the expression: 2(3x - 4y) + 5(x + 2y) - 7x

Steps:

1. Apply the distributive property to both parentheses: 2(3x - 4y) = 6x - 8y 5(x + 2y) = 5x + 10y
2. Rewrite the expression: 6x - 8y + 5x + 10y - 7x
3. Combine like terms: 6x + 5x - 7x = 4x -8y + 10y = 2y
4. Write the simplified expression: 4x + 2y

Example 4: Expressions with Exponents

Simplify the expression: 3x² + 5x - 2x² + 7x + 1

Steps:

1. Identify like terms: 3x² and -2x² are like terms; 5x and 7x are like terms; 1 is a constant term.
2. Combine like terms: 3x² - 2x² = x² 5x + 7x = 12x
3. Write the simplified expression: x² + 12x + 1

Example 5: Expressions with Fractions

Simplify the expression: (1/2)x + (2/3)x - 5

Steps:

1. Find a common denominator for the fractions: The common denominator for 2 and 3 is 6.
2. Rewrite the fractions with the common denominator: (1/2)x = (3/6)x (2/3)x = (4/6)x
3. Combine the like terms: (3/6)x + (4/6)x = (7/6)x
4. Write the simplified expression: (7/6)x - 5

Example 6: Expressions with Parentheses and Exponents

Simplify the expression: 2(x² + 3x) - 3(x² - 2x + 1)

Steps:

1. Apply the distributive property: 2(x² + 3x) = 2x² + 6x 3(x² - 2x + 1) = 3x² - 6x + 3
2. Rewrite the expression: 2x² + 6x - (3x² - 6x + 3) Remember to distribute the negative sign!
3. Combine like terms: 2x² - 3x² = -x² 6x + 6x = 12x
4. Write the simplified expression: -x² + 12x - 3

Dealing with Negative Signs and Parentheses

Remember that a negative sign in front of a parenthesis means you need to distribute the -1 to every term inside the parenthesis. This is often a source of errors, so pay close attention!

Scientific Explanation: The Underlying Principles

Simplifying algebraic expressions is based on fundamental algebraic properties:

• Commutative Property: The order of addition or multiplication doesn't change the result (a + b = b + a; ab = ba).
• Associative Property: The grouping of terms in addition or multiplication doesn't change the result ((a + b) + c = a + (b + c); (ab)c = a(bc)).
• Distributive Property: As discussed above, this property allows us to expand or factor expressions.
• Identity Property: Adding 0 or multiplying by 1 doesn't change the value (a + 0 = a; 1 * a = a).
• Inverse Property: Adding the opposite or multiplying by the reciprocal results in 0 or 1 (a + (-a) = 0; a * (1/a) = 1).

These properties are the foundation of all algebraic manipulation, including simplification.

Frequently Asked Questions (FAQ)

Q1: What happens if I have variables with different exponents?

A1: You can only combine like terms. Terms with different exponents (e.g., x² and x) cannot be combined. The simplified expression will have those terms separately.

Q2: Can I simplify expressions with radicals or roots?

A2: Yes, but the techniques are slightly more advanced. You'll often need to use rules of exponents and sometimes factoring to simplify expressions involving radicals.

Q3: What if I have fractions with variables in the denominator?

A3: These expressions require careful attention to rules of fractions and may involve techniques like finding a common denominator or simplifying complex fractions.

Q4: How do I check if my simplified expression is correct?

A4: You can substitute specific values for the variables into both the original and simplified expressions. If they yield the same result, your simplification is likely correct. However, this isn't a foolproof method as it only checks for one specific set of values.

Conclusion: Mastering the Art of Simplification

Simplifying algebraic expressions is a cornerstone of algebra. By understanding like terms, the distributive property, and applying the order of operations meticulously, you can master this essential skill. Practice is key—the more you work through different examples, the more confident and efficient you'll become. Remember to break down complex expressions into smaller, manageable steps, and always double-check your work to avoid common errors. With consistent effort, you'll find that simplifying algebraic expressions becomes second nature, paving the way for success in more advanced mathematical concepts.