What Is The Simplified Form Of The Following Expression

Simplifying Expressions: A Comprehensive Guide

Simplifying mathematical expressions is a fundamental skill in algebra and beyond. It involves manipulating an expression to make it easier to understand and work with, without changing its value. This guide will explore various techniques for simplifying expressions, focusing on different types of expressions and offering numerous examples to solidify your understanding. We'll cover simplifying expressions with integers, fractions, radicals, and exponents, providing a robust foundation for more advanced mathematical concepts.

Understanding the Basics: What Does "Simplifying" Mean?

Before diving into specific techniques, let's clarify what simplification entails. Essentially, we aim to rewrite an expression in its most concise and manageable form. This often involves:

• Combining like terms: Gathering terms with the same variables raised to the same powers.
• Removing parentheses: Using the distributive property to eliminate parentheses.
• Reducing fractions: Simplifying fractions by dividing the numerator and denominator by their greatest common divisor (GCD).
• Simplifying radicals: Removing perfect squares or other perfect powers from under the radical sign.
• Simplifying exponents: Using exponent rules to combine or simplify terms with exponents.

Simplifying Expressions with Integers

Simplifying expressions involving integers relies heavily on the order of operations (PEMDAS/BODMAS), which dictates the sequence of calculations:

1. Parentheses/Brackets: Evaluate expressions within parentheses first.
2. Exponents/Orders: Calculate exponents.
3. Multiplication and Division: Perform multiplication and division from left to right.
4. Addition and Subtraction: Perform addition and subtraction from left to right.

Example 1: Simplify 3 + 5 × 2 - 4 ÷ 2

Following PEMDAS:

1. Multiplication: 5 × 2 = 10
2. Division: 4 ÷ 2 = 2
3. Addition and Subtraction: 3 + 10 - 2 = 11

Therefore, the simplified expression is 11.

Simplifying Expressions with Fractions

Simplifying fractional expressions involves several steps:

1. Finding a Common Denominator: When adding or subtracting fractions, find the least common denominator (LCD) and rewrite each fraction with the LCD.
2. Combining Numerators: Add or subtract the numerators while keeping the common denominator.
3. Simplifying the Result: Reduce the resulting fraction to its lowest terms by dividing the numerator and denominator by their GCD.

Example 2: Simplify 1/2 + 2/3 - 1/6

1. Find the LCD: The LCD of 2, 3, and 6 is 6.
2. Rewrite Fractions: 1/2 = 3/6, 2/3 = 4/6
3. Combine Numerators: 3/6 + 4/6 - 1/6 = 6/6
4. Simplify: 6/6 = 1

Therefore, the simplified expression is 1.

Simplifying Expressions with Radicals

Simplifying radical expressions involves removing perfect squares (or higher powers) from under the radical sign.

Example 3: Simplify √72

1. Find Perfect Squares: 72 = 36 × 2, and 36 is a perfect square (6²).
2. Rewrite the Expression: √72 = √(36 × 2) = √36 × √2
3. Simplify: √36 = 6, so √72 simplifies to 6√2

Example 4: Simplify √(12x³y⁴)

1. Find Perfect Squares: 12 = 4 × 3, x³ = x² × x, y⁴ = (y²)²
2. Rewrite and Simplify: √(12x³y⁴) = √(4 × 3 × x² × x × (y²)²) = √4 × √x² × √(y²)² × √(3x) = 2xy²√(3x)

Simplifying Expressions with Exponents

Simplifying expressions with exponents utilizes various exponent rules:

• Product Rule: aᵐ × aⁿ = aᵐ⁺ⁿ
• Quotient Rule: aᵐ ÷ aⁿ = aᵐ⁻ⁿ
• Power Rule: (aᵐ)ⁿ = aᵐⁿ
• Power of a Product: (ab)ⁿ = aⁿbⁿ
• Power of a Quotient: (a/b)ⁿ = aⁿ/bⁿ
• Zero Exponent: a⁰ = 1 (a ≠ 0)
• Negative Exponent: a⁻ⁿ = 1/aⁿ

Example 5: Simplify (2x³y²)² × (3xy⁴)³

1. Apply Power Rules: (2x³y²)² = 4x⁶y⁴ and (3xy⁴)³ = 27x³y¹²
2. Combine using the Product Rule: 4x⁶y⁴ × 27x³y¹² = 108x⁹y¹⁶

Example 6: Simplify (x⁵y⁻²)/(x²y⁴)

1. Apply the Quotient Rule: x⁵/x² = x⁵⁻² = x³ and y⁻²/y⁴ = y⁻²⁻⁴ = y⁻⁶
2. Rewrite with Positive Exponents: x³y⁻⁶ = x³/y⁶

Simplifying Expressions with Parentheses

Removing parentheses often involves using the distributive property: a(b + c) = ab + ac.

Example 7: Simplify 3(x + 2) - 2(x - 1)

1. Distribute: 3(x + 2) = 3x + 6 and -2(x - 1) = -2x + 2
2. Combine Like Terms: 3x + 6 - 2x + 2 = x + 8

Combining Multiple Techniques

Often, simplifying expressions requires combining several of the techniques discussed above.

Example 8: Simplify (√8 + √18)²

1. Simplify Radicals: √8 = √(4 × 2) = 2√2 and √18 = √(9 × 2) = 3√2
2. Substitute and Simplify: (2√2 + 3√2)² = (5√2)² = 25 × 2 = 50

Example 9: Simplify (2x + 3)(x - 5) + 4x

1. FOIL (First, Outer, Inner, Last): (2x + 3)(x - 5) = 2x² - 10x + 3x - 15 = 2x² - 7x - 15
2. Combine with 4x: 2x² - 7x - 15 + 4x = 2x² - 3x - 15

Advanced Simplification Techniques

More advanced simplification techniques may involve factoring, completing the square, or using other algebraic manipulations depending on the complexity of the expression. These techniques are best explored in more advanced algebra courses.

Conclusion

Simplifying expressions is a crucial skill in mathematics. By mastering the techniques outlined in this guide, you can confidently tackle a wide range of expressions, transforming complex forms into their most concise and manageable equivalents. Remember to consistently apply the order of operations and utilize appropriate exponent and radical rules to achieve accurate simplification. Practice is key to developing fluency and understanding in this fundamental area of mathematics. The more you practice, the faster and more accurately you'll be able to simplify even the most challenging expressions.