Evaluate (2^2 X^2 Y / X Y^3)^2 For X=4 And Y=2 - A Step-by-Step Guide

Introduction to Evaluating Algebraic Expressions

In the realm of mathematics, evaluating algebraic expressions is a fundamental skill. It forms the cornerstone of more advanced concepts in algebra and calculus. Algebraic expressions are mathematical phrases that combine numbers, variables, and mathematical operations. Variables are symbols, usually letters, that represent unknown quantities. Our task involves substituting specific values for these variables and then simplifying the expression to arrive at a numerical answer. This process is not merely about arriving at a number; it's about understanding the structure of the expression and how different operations interact. The ability to evaluate expressions accurately is crucial in various fields, including engineering, physics, and computer science, where mathematical models are used to represent real-world phenomena. In this article, we will delve into a specific example, showcasing the step-by-step process of evaluating an algebraic expression. This will not only demonstrate the practical application of algebraic principles but also highlight the importance of careful arithmetic and attention to detail. Mastering the evaluation of expressions is like learning the alphabet of mathematics – it's the basic building block upon which more complex ideas are built. So, let's embark on this journey, unravel the expression, and discover the numerical value it holds when specific values are assigned to its variables.

Problem Statement: Evaluating the Given Expression

In this article, we aim to evaluate the algebraic expression $\left(\frac{2^2 x^2 y}{x y^3}\right)^2$ for the specific values of $x = 4$ and $y = 2$. This problem serves as an excellent example to illustrate the process of simplifying and evaluating expressions. The expression involves several key components: numerical coefficients, variables raised to powers, and a combination of multiplication and division operations. The presence of exponents, both inside and outside the parentheses, adds another layer of complexity, requiring a careful application of the order of operations. Our task is to substitute the given values of $x$ and $y$ into the expression and then simplify it step by step to arrive at a final numerical answer. This process will involve several stages, including simplifying the expression within the parentheses, applying the exponent outside the parentheses, and finally performing the arithmetic calculations. The importance of this problem lies not just in finding the solution but in understanding the methodology behind it. By breaking down the expression into smaller, manageable parts, we can avoid errors and gain a deeper understanding of algebraic manipulation. This problem is representative of the kind of algebraic challenges encountered in various mathematical contexts, making its solution a valuable exercise in mathematical reasoning and problem-solving.

Step 1: Simplify the Expression Inside the Parentheses

The first crucial step in evaluating the expression $\left(\frac{2^2 x^2 y}{x y^3}\right)^2$ is to simplify the terms inside the parentheses. This involves simplifying both the numerical coefficients and the variable terms. Let's begin by addressing the numerical coefficient $2^2$. This term means 2 multiplied by itself, which equals 4. So, we can replace $2^2$ with 4 in our expression. Next, we focus on simplifying the variable terms. We have $x^2$ in the numerator and $x$ in the denominator. When dividing terms with the same base, we subtract the exponents. Thus, $x^2$ divided by $x$ simplifies to $x^{2-1}$, which is simply $x$. Similarly, we have $y$ in the numerator and $y^3$ in the denominator. Dividing $y$ by $y^3$ gives us $y^{1-3}$, which simplifies to $y^{-2}$. It's important to remember that a negative exponent means we take the reciprocal of the base raised to the positive exponent. So, $y^{-2}$ is the same as $\frac{1}{y^2}$. Putting it all together, the expression inside the parentheses simplifies to $\frac{4x}{y^2}$. This simplification is a critical step as it reduces the complexity of the expression, making it easier to substitute the given values and proceed with the calculation. The ability to simplify expressions like this is a fundamental skill in algebra, enabling us to tackle more complex problems with confidence. By breaking down the expression and applying the rules of exponents, we've made significant progress towards finding the final solution.

Step 2: Apply the Exponent Outside the Parentheses

Having simplified the expression inside the parentheses to $\frac{4x}{y^2}$, the next step is to apply the exponent of 2 that is outside the parentheses. This means we need to square the entire fraction, which involves squaring both the numerator and the denominator separately. Squaring the numerator, $(4x)^2$, means multiplying $4x$ by itself. This can be written as $(4x) \times (4x)$. Using the rules of exponents, we multiply the coefficients (4 and 4) to get 16, and we square the variable $x$ to get $x^2$. So, $(4x)^2$ simplifies to $16x^2$. Now, let's consider the denominator, $(y^2)^2$. When we raise a power to another power, we multiply the exponents. Therefore, $(y^2)^2$ simplifies to $y^{2 \times 2}$, which is $y^4$. Putting the simplified numerator and denominator together, we have the expression $\frac{16x^2}{y^4}$. This step is crucial because it eliminates the parentheses and further simplifies the expression, making it ready for the substitution of the given values of $x$ and $y$. Understanding how to apply exponents to expressions involving fractions and variables is a key skill in algebra. It allows us to manipulate expressions and transform them into simpler forms, which is essential for solving equations and tackling more complex mathematical problems. By carefully applying the rules of exponents, we've successfully simplified the expression and are now one step closer to finding the final numerical value.

Step 3: Substitute the Values of x and y

With the expression simplified to $\frac{16x^2}{y^4}$, we are now ready to substitute the given values of $x$ and $y$. We are given that $x = 4$ and $y = 2$. Substituting these values into the expression, we replace $x$ with 4 and $y$ with 2. This gives us $\frac{16(4)^2}{(2)^4}$. It's crucial to perform the substitution carefully, ensuring that each variable is replaced with its corresponding value. This step bridges the gap between the algebraic form of the expression and its numerical evaluation. The act of substitution is a fundamental concept in algebra, allowing us to find the value of an expression for specific instances. It's like plugging in the coordinates of a point into an equation to see if it lies on the line or curve represented by that equation. In this case, we are plugging in the values of $x$ and $y$ to find the numerical value of the expression under these conditions. This step is straightforward but requires precision to avoid errors. By correctly substituting the values, we set the stage for the final arithmetic calculations that will lead us to the solution. Now that we have the expression in numerical form, the next step is to perform the operations according to the order of operations to arrive at the final answer.

Step 4: Perform the Arithmetic Calculations

Having substituted the values of $x$ and $y$ into the expression, we now have $\frac{16(4)^2}{(2)^4}$. The next step is to perform the arithmetic calculations to arrive at a numerical answer. Following the order of operations, we first address the exponents. We have $4^2$ in the numerator, which means 4 multiplied by itself, resulting in 16. So, we replace $4^2$ with 16 in the expression. Similarly, we have $2^4$ in the denominator, which means 2 multiplied by itself four times, resulting in 16. So, we replace $2^4$ with 16. Now our expression looks like $\frac{16(16)}{16}$. Next, we perform the multiplication in the numerator. 16 multiplied by 16 equals 256. So, the expression becomes $\frac{256}{16}$. Finally, we perform the division. 256 divided by 16 equals 16. Therefore, the final numerical value of the expression is 16. This step demonstrates the importance of following the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). By performing the calculations in the correct order, we ensure that we arrive at the correct answer. This arithmetic calculation is the culmination of all the previous steps, bringing us to the solution of the problem. The ability to perform these calculations accurately is a crucial skill in mathematics, and it's the final piece of the puzzle in evaluating the given expression.

Final Answer: The Result of the Evaluation

After carefully simplifying the expression $\left(\frac{2^2 x^2 y}{x y^3}\right)^2$ and substituting the given values of $x = 4$ and $y = 2$, we have arrived at the final answer. By following the step-by-step process of simplifying the expression inside the parentheses, applying the exponent outside the parentheses, substituting the values of $x$ and $y$, and performing the arithmetic calculations, we have determined that the value of the expression is 16. This result is the culmination of all the steps taken, demonstrating the power of algebraic manipulation and the importance of careful arithmetic. The final answer of 16 represents the numerical value of the expression under the specified conditions. It's a single, concrete number that encapsulates the result of all the operations performed. This process of evaluating expressions is a fundamental skill in mathematics, and it's essential for solving a wide range of problems in various fields. The ability to accurately evaluate expressions is a testament to one's understanding of algebraic principles and arithmetic operations. In conclusion, the value of the expression $\left(\frac{2^2 x^2 y}{x y^3}\right)^2$ when $x = 4$ and $y = 2$ is 16. This completes our evaluation and provides a clear and concise answer to the problem.