Simplifying Expressions An In-Depth Look At (4x³y⁵)(3x⁵y)²
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In the realm of algebra, simplifying expressions is a fundamental skill. It allows us to represent complex mathematical relationships in a more concise and understandable manner. This article delves into the process of simplifying a given algebraic expression, focusing on the application of exponent rules and the distributive property. We will meticulously break down each step, providing a comprehensive understanding of how to arrive at the equivalent simplified form. Our goal is to transform the initial expression, which may appear daunting at first glance, into a more manageable and transparent representation. This involves not only simplifying the expression but also ensuring that each step is clear and logical, thereby enhancing your grasp of algebraic manipulation.
Deconstructing the Expression: A Step-by-Step Simplification
The Initial Expression: (4x³y⁵)(3x⁵y)²
At the heart of our exploration is the expression (4x³y⁵)(3x⁵y)². This expression combines coefficients, variables, and exponents, presenting an opportunity to apply various algebraic principles. To simplify this, we'll need to understand how to handle exponents both inside and outside parentheses, as well as how to multiply terms with the same base. The process begins with a careful examination of the expression's structure, identifying the order of operations and the relevant exponent rules. Understanding the nuances of this expression is crucial for mastering the art of algebraic simplification.
Step 1: Applying the Power of a Product Rule
The first key step in simplifying this expression is to address the exponent outside the parentheses in the second term. Specifically, we have (3x⁵y)². The power of a product rule states that (ab)ⁿ = aⁿbⁿ. Applying this rule, we distribute the exponent 2 to each factor within the parentheses. This means that 3 is raised to the power of 2, x⁵ is raised to the power of 2, and y is raised to the power of 2. This transformation is crucial as it allows us to separate the terms and deal with the exponents individually, making the simplification process more manageable.
This yields: (3x⁵y)² = 3² * (x⁵)² * y²
Step 2: Simplifying the Exponents
Continuing our journey towards simplification, we now focus on the exponents themselves. 3² is straightforward and equals 9. For (x⁵)², we use the power of a power rule, which states that (aᵐ)ⁿ = aᵐⁿ. This means we multiply the exponents: 5 * 2 = 10. So, (x⁵)² simplifies to x¹⁰. For y², there's no further simplification needed as it's already in its simplest form. This step is a pivotal moment in the simplification process, as it directly reduces the complexity of the exponential terms.
Therefore, we have: 3² * (x⁵)² * y² = 9x¹⁰y²
Step 3: Rewriting the Expression
Now that we've simplified the second term, we rewrite the entire expression, replacing (3x⁵y)² with its simplified form, 9x¹⁰y². This step is crucial for bringing together all the simplified components, setting the stage for the final simplification through multiplication. It allows us to view the expression in a more consolidated form, making the subsequent steps clearer and more intuitive.
This gives us: (4x³y⁵)(9x¹⁰y²)
Step 4: Applying the Product of Powers Rule
The next critical step involves multiplying the two terms together. We multiply the coefficients (4 and 9) and then apply the product of powers rule to the variables. The product of powers rule states that aᵐ * aⁿ = aᵐ+ⁿ. This means we add the exponents of like bases. For x, we have x³ * x¹⁰, which becomes x³⁺¹⁰ = x¹³. For y, we have y⁵ * y², which becomes y⁵⁺² = y⁷. This step is the heart of the simplification, where we combine terms and reduce the expression to its most basic form.
So, multiplying the terms, we get: 4 * 9 * x³ * x¹⁰ * y⁵ * y² = 36x¹³y⁷
Step 5: The Final Simplified Expression
Finally, we arrive at the fully simplified expression. By multiplying the coefficients and adding the exponents of like variables, we've transformed the original complex expression into a concise and easily understandable form. The simplified expression is 36x¹³y⁷. This is the culmination of our step-by-step simplification process, showcasing the power of algebraic rules in reducing complexity.
Thus, the equivalent expression is: 36x¹³y⁷
Why is Simplifying Expressions Important?
Simplifying expressions is not just a mathematical exercise; it's a critical skill with real-world applications. In mathematics, simplified expressions are easier to work with in further calculations, such as solving equations or graphing functions. Beyond mathematics, simplification is essential in fields like physics, engineering, and computer science. For instance, in physics, simplifying equations can help predict the behavior of physical systems. In computer science, it can optimize algorithms and improve performance. Simplifying expressions also enhances clarity and understanding, allowing for easier communication of mathematical ideas. By reducing complexity, we make problems more accessible and solutions more apparent.
Common Mistakes to Avoid When Simplifying Expressions
When simplifying expressions, it's crucial to avoid common pitfalls that can lead to incorrect results. One frequent mistake is incorrectly applying the order of operations (PEMDAS/BODMAS). Exponents should be handled before multiplication or division, and addition or subtraction should be done last. Another common error is misapplying exponent rules, such as forgetting to distribute an exponent to all factors within parentheses. Additionally, students often make mistakes when combining like terms, especially when dealing with negative coefficients or exponents. Careful attention to detail and a thorough understanding of algebraic rules are essential to avoid these errors and achieve accurate simplifications. Double-checking each step and practicing regularly can significantly reduce the likelihood of mistakes.
Practice Problems: Sharpening Your Skills
To solidify your understanding of simplifying expressions, practice is essential. Try simplifying the following expressions:
- (2a⁴b²)(5a²b³)²
- (3x²y)(4x³y⁴)³
- (7m³n⁵)²(2mn²)
Work through each problem step by step, paying close attention to the exponent rules and the order of operations. Check your answers with a peer or consult a textbook to ensure accuracy. The more you practice, the more confident and proficient you'll become in simplifying expressions. Practice not only reinforces the concepts but also helps you develop problem-solving skills and mathematical intuition.
Conclusion: Mastering Algebraic Simplification
In conclusion, simplifying algebraic expressions is a fundamental skill in mathematics with broad applications. Through a step-by-step approach, we've demonstrated how to simplify the expression (4x³y⁵)(3x⁵y)², arriving at the equivalent form of 36x¹³y⁷. We've emphasized the importance of understanding and applying exponent rules, such as the power of a product rule and the product of powers rule. Avoiding common mistakes and practicing regularly are key to mastering this skill. By simplifying expressions, we not only make mathematical problems more manageable but also enhance our overall understanding of algebraic concepts. This skill is a cornerstone for success in higher-level mathematics and various scientific and technical fields.
