Simplifying Algebraic Expressions An In-Depth Guide To (2/3)a⁴b⁴ × (-3a²b) ÷ (1/4)a⁵b⁷

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Algebraic expressions are the foundation of mathematics, and simplifying them is a crucial skill for success in algebra and beyond. This article provides a detailed, step-by-step guide on how to simplify a complex algebraic expression, specifically focusing on the expression (2/3)a⁴b⁴ × (-3a²b) ÷ (1/4)a⁵b⁷. We will break down each step, explaining the underlying principles and rules of exponents, ensuring a clear understanding of the simplification process. Mastering the simplification of algebraic expressions like this one will empower you to tackle more complex mathematical problems with confidence.

Understanding the Fundamentals of Algebraic Simplification

Before diving into the specific expression, let's first establish the fundamental principles that govern algebraic simplification. Algebraic simplification involves reducing an expression to its simplest form while maintaining its mathematical equivalence. This often involves combining like terms, applying the order of operations (PEMDAS/BODMAS), and utilizing the rules of exponents. Understanding these basic principles is key to correctly and efficiently simplifying any algebraic expression.

Key Concepts to Remember:

  • Order of Operations: This dictates the sequence in which operations are performed: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Commonly remembered by the acronyms PEMDAS or BODMAS.
  • Combining Like Terms: Like terms are those that have the same variables raised to the same powers. For example, 3x² and -5x² are like terms, but 3x² and 3x³ are not. We can combine like terms by adding or subtracting their coefficients.
  • Rules of Exponents: These rules are crucial for simplifying expressions involving powers. Some key rules include:
    • Product of Powers: xᵐ * xⁿ = xᵐ⁺ⁿ
    • Quotient of Powers: xᵐ / xⁿ = xᵐ⁻ⁿ
    • Power of a Power: (xᵐ)ⁿ = xᵐⁿ
    • Power of a Product: (xy)ⁿ = xⁿyⁿ
    • Power of a Quotient: (x/y)ⁿ = xⁿ/yⁿ
    • Negative Exponent: x⁻ⁿ = 1/xⁿ
    • Zero Exponent: x⁰ = 1 (where x ≠ 0)

With these fundamentals in mind, we are now well-equipped to tackle the given expression. We'll apply these principles systematically to arrive at the simplified form.

Step-by-Step Simplification of (2/3)a⁴b⁴ × (-3a²b) ÷ (1/4)a⁵b⁷

Now, let's break down the simplification process of the expression (2/3)a⁴b⁴ × (-3a²b) ÷ (1/4)a⁵b⁷ step-by-step. We'll meticulously apply the order of operations and the rules of exponents to arrive at the final simplified form. Understanding each step is paramount for mastering algebraic simplification.

Step 1: Multiply the First Two Terms

The first step, according to the order of operations, is to perform the multiplication. We need to multiply (2/3)a⁴b⁴ by (-3a²b). To do this, we multiply the coefficients and then apply the product of powers rule for the variables.

(2/3)a⁴b⁴ × (-3a²b) = (2/3) × (-3) × a⁴ × a² × b⁴ × b¹

Let's break this down further:

  • Multiply the coefficients: (2/3) × (-3) = -2
  • Multiply the 'a' terms: a⁴ × a² = a⁴⁺² = a⁶ (Applying the product of powers rule)
  • Multiply the 'b' terms: b⁴ × b¹ = b⁴⁺¹ = b⁵ (Applying the product of powers rule)

Therefore, the result of the multiplication is: -2a⁶b⁵

This simplified expression now replaces the first two terms in the original expression, making it easier to proceed with the next operation.

Step 2: Divide by the Third Term

Next, we need to divide the result from Step 1, which is -2a⁶b⁵, by the third term, (1/4)a⁵b⁷. Division by a fraction is equivalent to multiplication by its reciprocal. Therefore, we can rewrite the division as multiplication.

-2a⁶b⁵ ÷ (1/4)a⁵b⁷ = -2a⁶b⁵ × (4/1)a⁻⁵b⁻⁷

Here, dividing by (1/4) is the same as multiplying by 4, and we've rewritten the division of the variable terms using negative exponents (a⁻⁵ and b⁻⁷) to prepare for applying the quotient of powers rule.

Now, we can multiply the coefficients and apply the rules of exponents for division:

  • Multiply the coefficients: -2 × 4 = -8
  • Divide the 'a' terms: a⁶ × a⁻⁵ = a⁶⁻⁵ = a¹ = a (Applying the product of powers rule with a negative exponent)
  • Divide the 'b' terms: b⁵ × b⁻⁷ = b⁵⁻⁷ = b⁻² (Applying the product of powers rule with a negative exponent)

Therefore, the result of the division is: -8ab⁻²

Step 3: Simplify the Negative Exponent

The final step is to simplify the expression by removing the negative exponent. Recall that a negative exponent indicates a reciprocal. Therefore, b⁻² is equivalent to 1/b².

-8ab⁻² = -8a × (1/b²) = -8a/b²

This step transforms the expression into its most simplified form, eliminating the negative exponent and presenting the final result as a fraction.

The Final Simplified Expression

After performing all the necessary steps, the simplified form of the original expression (2/3)a⁴b⁴ × (-3a²b) ÷ (1/4)a⁵b⁷ is:

-8a/b²

This is the most concise and simplified representation of the given algebraic expression. By following the order of operations and applying the rules of exponents, we have successfully reduced the complex expression to its simplest form.

Common Mistakes to Avoid When Simplifying Algebraic Expressions

Simplifying algebraic expressions can be tricky, and it's easy to make mistakes if you're not careful. Here are some common pitfalls to avoid:

  1. Ignoring the Order of Operations (PEMDAS/BODMAS): This is a fundamental error. Always ensure you perform operations in the correct order: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
  2. Incorrectly Combining Like Terms: Only terms with the same variable raised to the same power can be combined. For instance, 3x² and 5x are not like terms and cannot be combined.
  3. Misapplying the Rules of Exponents: A common mistake is to add exponents when multiplying terms with different bases or to multiply exponents when multiplying terms with the same base. Review and understand the rules of exponents thoroughly.
  4. Forgetting to Distribute: When multiplying a term by an expression in parentheses, ensure you distribute the term to every term inside the parentheses. For example, a(b + c) = ab + ac.
  5. Errors with Negative Signs: Be extremely careful when dealing with negative signs. A misplaced negative sign can completely change the answer. Pay close attention when multiplying or dividing by negative numbers.
  6. Incorrectly Simplifying Fractions: Ensure you simplify fractions completely by dividing both the numerator and denominator by their greatest common factor.
  7. Skipping Steps: While it may be tempting to rush through the process, skipping steps can lead to errors. Take your time and write out each step clearly, especially when dealing with complex expressions.

By being aware of these common mistakes, you can significantly improve your accuracy and confidence in simplifying algebraic expressions. Practice and attention to detail are key to mastering this skill.

Practice Problems to Sharpen Your Skills

To solidify your understanding of simplifying algebraic expressions, let's work through some practice problems. These examples will help you apply the concepts and techniques we've discussed and identify areas where you may need further practice.

Practice Problem 1: Simplify: (4x³y²) × (-2xy⁵) ÷ (6x²y³)

Solution:

  1. Multiply the first two terms: (4x³y²) × (-2xy⁵) = -8x⁴y⁷
  2. Divide by the third term: -8x⁴y⁷ ÷ (6x²y³) = (-8/6)x²y⁴ = (-4/3)x²y⁴
  3. Final Simplified Expression: (-4/3)x²y⁴

Practice Problem 2: Simplify: (1/2)a²b × (4ab³) ÷ (2/3)a³b²

Solution:

  1. Multiply the first two terms: (1/2)a²b × (4ab³) = 2a³b⁴
  2. Divide by the third term: 2a³b⁴ ÷ (2/3)a³b² = 2a³b⁴ × (3/2)a⁻³b⁻² = 3a⁰b²
  3. Simplify a⁰: 3a⁰b² = 3 × 1 × b² = 3b²
  4. Final Simplified Expression: 3b²

Practice Problem 3: Simplify: (-5p⁴q) × (2pq²) ÷ (1/5)p²q⁴

Solution:

  1. Multiply the first two terms: (-5p⁴q) × (2pq²) = -10p⁵q³
  2. Divide by the third term: -10p⁵q³ ÷ (1/5)p²q⁴ = -10p⁵q³ × 5p⁻²q⁻⁴ = -50p³q⁻¹
  3. Simplify the negative exponent: -50p³q⁻¹ = -50p³/q
  4. Final Simplified Expression: -50p³/q

By working through these practice problems, you'll gain confidence in your ability to simplify algebraic expressions. Remember to focus on each step, apply the rules correctly, and double-check your work.

Conclusion: Mastering Algebraic Simplification

In conclusion, simplifying algebraic expressions is a foundational skill in mathematics. By understanding the order of operations, mastering the rules of exponents, and practicing diligently, you can confidently tackle complex expressions and achieve accurate results. We've walked through a detailed example of simplifying (2/3)a⁴b⁴ × (-3a²b) ÷ (1/4)a⁵b⁷, highlighting each step and the underlying principles involved. Remember to avoid common mistakes, such as misapplying the order of operations or incorrectly combining like terms. The key to success lies in a systematic approach, attention to detail, and consistent practice.

By mastering these techniques, you'll not only excel in algebra but also build a strong foundation for more advanced mathematical concepts. So, keep practicing, keep learning, and embrace the power of algebraic simplification! This skill will undoubtedly serve you well in your mathematical journey and beyond.