Simplify The Expression \(\frac{x^2(y^3)^4}{xy^5}\) And Express It In The Form *x^a* *y^b*. Determine The Values Of *a* And *b*.

In the realm of mathematics, simplifying algebraic expressions is a fundamental skill. It's not just about making things look neater; it's about unveiling the underlying structure and relationships within the expression. This article delves into the process of simplifying the expression ${\frac{x^2(y^3)^4}{xy^5}}$ and determining the values of a and b when it is expressed in the form x^a y^b. We'll break down each step, providing a clear and concise explanation to enhance your understanding of exponent rules and algebraic manipulation. This skill is not just confined to textbooks; it's the cornerstone of various mathematical disciplines, including calculus, linear algebra, and differential equations. A firm grasp of simplifying algebraic expressions empowers you to tackle more complex problems with confidence and precision. Moreover, it fosters a deeper appreciation for the elegance and power of mathematical notation, allowing you to express intricate relationships in a compact and meaningful way. Remember, practice is key to mastering this skill. By working through numerous examples, you'll develop an intuitive understanding of how different operations interact, enabling you to simplify expressions efficiently and accurately. As you progress, you'll discover that simplifying expressions is not merely a mechanical process; it's an art form that requires a blend of logical reasoning and creative problem-solving. So, let's embark on this journey of simplification and uncover the beauty hidden within algebraic expressions.

Understanding the Problem

The problem presented involves simplifying an algebraic expression containing variables with exponents. We are given the expression ${\frac{x^2(y^3)^4}{xy^5}}$ and asked to rewrite it in the form x^a y^b. Our task is to find the numerical values of the exponents a and b. To accomplish this, we will leverage the fundamental rules of exponents, which govern how powers behave under various operations. These rules are the bedrock of algebraic simplification, providing a systematic way to manipulate expressions involving exponents. One of the key rules we'll employ is the power of a power rule, which states that (x^m)ⁿ = x^m*n. This rule allows us to simplify expressions where an exponent is raised to another power. Another crucial rule is the quotient of powers rule, which dictates that x^m / xⁿ = x^m-n. This rule helps us simplify expressions involving division of terms with the same base. Additionally, we'll utilize the product of powers rule, which states that x^m * xⁿ = x^m+n. This rule is essential for combining terms with the same base that are being multiplied. By applying these exponent rules in a strategic manner, we can systematically simplify the given expression and isolate the values of a and b. Remember, understanding the underlying principles behind these rules is just as important as memorizing them. This deeper understanding will enable you to adapt your approach to a wide range of algebraic simplification problems, fostering a more flexible and robust problem-solving skillset. So, let's dive into the simplification process and see how these rules come into play.

Step-by-Step Simplification

Let's break down the simplification process step-by-step:

1. Apply the Power of a Power Rule:

   • We start with the expression ${\frac{x^2(y^3)^4}{xy^5}}$. The first step is to simplify the term (y³)⁴. According to the power of a power rule, (x^m)ⁿ = x^m*n, we multiply the exponents: 3 * 4 = 12. So, (y³)⁴ simplifies to y¹². This transformation is crucial because it allows us to consolidate the exponents associated with the variable y. By reducing the complexity of the expression, we pave the way for subsequent simplifications. The expression now becomes ${\frac{x^2y^{12}}{xy^5}}$. This step highlights the importance of recognizing and applying the appropriate exponent rules in the correct order. Each rule acts as a tool in our simplification arsenal, and mastering their application is key to unlocking the solution. Remember, the power of a power rule is a fundamental concept in algebra, and its mastery is essential for tackling more complex expressions involving exponents.

2. Apply the Quotient of Powers Rule:

   • Now we have ${\frac{x^2y^{12}}{xy^5}}$. We can simplify the terms with the same base by applying the quotient of powers rule, x^m / xⁿ = x^m-n. For the x terms, we have x² / x¹ (remember that x is the same as x¹), which simplifies to x^2-1 = x¹ = x. For the y terms, we have y¹² / y⁵, which simplifies to y^12-5 = y⁷. This step demonstrates the power of the quotient of powers rule in streamlining expressions involving division. By subtracting the exponents of terms with the same base, we effectively cancel out common factors, leading to a more concise representation. The application of this rule not only simplifies the expression but also reveals the underlying relationships between the variables. It's a testament to the elegance and efficiency of mathematical notation, where complex operations can be represented in a compact and meaningful way. The expression now becomes xy⁷, a significantly simpler form than our starting point.

3. Combine the Simplified Terms:

   • After applying the quotient of powers rule, we have xy⁷. This expression is now in the desired form of x^a y^b. By comparing the simplified expression with the target form, we can directly identify the values of a and b. In this case, the exponent of x is 1 (since x is the same as x¹), so a = 1. The exponent of y is 7, so b = 7. This final step underscores the importance of recognizing patterns and making connections between different representations of the same mathematical object. By expressing the simplified expression in the desired form, we can readily extract the values of the unknowns, providing a clear and concise solution to the problem. The ability to move seamlessly between different representations is a hallmark of mathematical fluency, and it's a skill that will serve you well in a variety of mathematical contexts. Therefore, the simplified expression xy⁷ directly yields the solution to our problem, highlighting the effectiveness of the step-by-step simplification process.

Finding the Values of a and b

Based on the simplified expression, we can now easily determine the values of a and b. The simplified expression is xy⁷, which can be written as x¹ y⁷. Comparing this to the form x^a y^b, we can see that:

• a = 1
• b = 7

Therefore, the values of a and b are 1 and 7, respectively. This final step is the culmination of our simplification journey, providing a clear and definitive answer to the problem. By carefully applying the exponent rules and systematically simplifying the expression, we have successfully unveiled the values of a and b. This process demonstrates the power of algebraic manipulation in revealing the underlying structure of mathematical expressions. It also reinforces the importance of attention to detail and accuracy in mathematical calculations. A single error in applying the exponent rules can lead to an incorrect result. Therefore, it's crucial to double-check each step and ensure that the operations are performed correctly. With practice and diligence, you can develop a strong command of algebraic simplification techniques, enabling you to tackle a wide range of mathematical problems with confidence and precision. The values a = 1 and b = 7 represent the solution to the problem, showcasing the effectiveness of our step-by-step approach.

Conclusion

In conclusion, by applying the rules of exponents, we successfully simplified the expression ${\frac{x^2(y^3)^4}{xy^5}}$ to xy⁷. This allowed us to determine that a = 1 and b = 7. This exercise demonstrates the importance of understanding and applying exponent rules in simplifying algebraic expressions. The ability to simplify expressions is a fundamental skill in algebra and is crucial for solving more complex mathematical problems. Throughout this article, we have emphasized the importance of a step-by-step approach, highlighting the role of each exponent rule in the simplification process. By mastering these rules, you can confidently tackle a wide range of algebraic expressions, unlocking their underlying structure and relationships. Remember, practice is key to developing proficiency in simplifying expressions. The more you work through examples, the more intuitive the process will become. As you gain experience, you'll also develop a deeper appreciation for the elegance and power of algebraic notation. Simplifying expressions is not just about manipulating symbols; it's about revealing the hidden beauty and order within mathematical relationships. The values a = 1 and b = 7 represent the culmination of our simplification journey, providing a clear and concise solution to the problem. This success underscores the effectiveness of the exponent rules and the importance of a systematic approach in algebraic manipulation. So, continue to practice and explore the world of algebraic expressions, and you'll discover the power and beauty that lies within.