Hey guys! Today, we're diving into a fun math problem where we'll evaluate the expression (2^2 x^2 / x y2)2 given the values of x and y. Don't worry, it sounds more complicated than it actually is. We'll break it down step-by-step, so you can follow along easily. So, let's get started and see how to solve this! This kind of problem is a fantastic way to sharpen your algebra skills and get comfortable with exponents and fractions. Remember, the key is to take it one step at a time, and you'll be a pro in no time. Understanding how to evaluate expressions like this is crucial not just for math class, but also for various real-world applications, such as calculating areas, volumes, and even financial growth. So, stick with me, and let's tackle this problem together! We'll explore the fundamental principles behind simplifying algebraic expressions and substituting values, which are essential skills for any math enthusiast. By the end of this article, you'll not only know the answer to this specific problem but also gain the confidence to approach similar challenges. We'll also touch upon common mistakes to avoid and strategies to double-check your work. So, let's embark on this mathematical journey and unravel the mysteries of this expression! We'll transform what seems like a daunting task into a clear and manageable process, equipping you with the tools to excel in algebra. Remember, math is not just about finding the right answer; it's about understanding the process and developing problem-solving skills that will serve you well in many areas of life. So, let's make math fun and engaging, and let's get started!
Step 1: Simplify the Expression
First things first, let's simplify the expression inside the parentheses before we plug in any numbers. This will make our lives much easier. We have (2^2 x^2 / x y2)2. Remember the order of operations (PEMDAS/BODMAS)? We'll be using that quite a bit here. When we look at our expression, we see that we can simplify the terms inside the parentheses before we deal with the exponent outside. This is a crucial step because simplifying first reduces the chances of making errors later on. Think of it like organizing your workspace before you start a big project – it makes the whole process smoother and more efficient. By simplifying within the parentheses, we're essentially cleaning up the expression, making it less cluttered and easier to handle. This strategy is particularly useful when dealing with complex expressions involving multiple variables and exponents. It's like taking a tangled mess of wires and neatly organizing them so you can see what's connected to what. So, let's focus on simplifying the terms inside the parentheses, and we'll see how much easier the overall problem becomes. We'll be using basic algebraic principles, such as the rules of exponents and division, to streamline the expression. This not only makes the calculations simpler but also helps us understand the structure of the expression better. So, let's dive in and start simplifying!
Let's start by simplifying 2^2, which is simply 2 times 2, giving us 4. So our expression now looks like (4x^2 / x y2)2. Now, we can see that we have x^2 in the numerator and x in the denominator. Remember that when we divide terms with the same base, we subtract the exponents. So, x^2 / x simplifies to x^(2-1) which is just x. This is a fundamental rule of exponents that we'll be using throughout this problem. It's like saying, "How many x's are left when we cancel out one x from x squared?" The answer is, of course, one x. Understanding this rule is crucial for simplifying algebraic expressions efficiently. It's a powerful tool that allows us to condense terms and make equations more manageable. Think of it like simplifying a fraction – we're essentially reducing the expression to its simplest form. This step is essential because it allows us to work with smaller, more manageable numbers and variables, making the overall calculation less prone to errors. So, we've simplified the x terms, and now we're one step closer to having a fully simplified expression within the parentheses. Remember, each simplification we make brings us closer to the final solution, so let's keep going and tackle the remaining terms! By breaking down the expression into smaller, more manageable parts, we can apply the rules of algebra more effectively and arrive at the correct answer with confidence.
Our expression is now (4x / y2)2. This looks much simpler, right? We've successfully simplified the terms inside the parentheses as much as possible. Now we need to deal with the exponent outside the parentheses. Remember, when we raise a fraction to a power, we raise both the numerator and the denominator to that power. This is a key concept in dealing with exponents and fractions. It's like distributing the exponent to each term within the parentheses. So, (4x / y2)2 becomes (4x)^2 / (y2)2. We're essentially applying the exponent to each part of the fraction, ensuring that we account for its effect on the entire expression. This step is crucial because it allows us to further simplify the expression and prepare it for the substitution of values. Think of it like expanding a bracket – we're applying the exponent to each term within the parentheses, just as we would multiply each term inside a bracket by a factor outside. This rule is fundamental in algebra and is used extensively in various mathematical contexts. So, now we've distributed the exponent, and we're ready to simplify further. Remember, each step we take brings us closer to the final solution, and by understanding these fundamental rules, we're building a solid foundation for tackling more complex problems. Let's keep going and see what we can simplify next!
Now let's apply the exponent to each term. (4x)^2 means 4 squared times x squared, which is 16x^2. And (y2)2 means y squared raised to the power of 2. When we raise a power to another power, we multiply the exponents. So, (y2)2 becomes y^(22)*, which is y^4. This is another important rule of exponents that's crucial for simplifying expressions. It's like saying, "If we square y squared, what do we get?" The answer is y to the power of 4. Understanding this rule allows us to handle expressions with nested exponents efficiently. Think of it like unpacking a box within a box – we're applying the exponent to each layer. This step is essential because it allows us to fully expand the expression and prepare it for the final substitution. We've now simplified the numerator and the denominator separately, making the overall expression much clearer. Remember, each simplification we make is a step towards the final answer, and by mastering these exponent rules, we're building a strong foundation for more advanced mathematical concepts. So, let's keep going and see how we can further refine our expression and eventually substitute the given values.
So, our simplified expression is now 16x^2 / y^4. Great job! We've done the hard part of simplifying the expression. Now comes the fun part – plugging in the values for x and y.
Step 2: Substitute the Values
We're given that x = 3 and y = 2. So, let's substitute these values into our simplified expression 16x^2 / y^4. This is a straightforward step where we replace the variables with their corresponding numerical values. It's like filling in the blanks in a sentence – we're replacing the placeholders with the actual values. This step is crucial because it allows us to move from an algebraic expression to a numerical value. Think of it like translating a recipe into a finished dish – we're taking the abstract ingredients and turning them into something concrete. Substituting values is a fundamental skill in algebra and is used extensively in various mathematical applications. It allows us to evaluate expressions and equations for specific scenarios. So, let's carefully substitute the values and see what numerical expression we get. Remember, accuracy is key in this step, so let's double-check that we're substituting the correct values for the correct variables. By substituting the values, we're essentially bringing the expression to life, giving it a tangible numerical value. Let's keep going and see what the final answer is!
Replacing x with 3, we get 16 * (3^2) in the numerator. Replacing y with 2, we get 2^4 in the denominator. So, our expression now looks like (16 * 3^2) / 2^4. We've successfully substituted the values, and now we're ready to perform the calculations. This step is crucial because it transforms the algebraic expression into a purely numerical one, which we can then evaluate using basic arithmetic operations. Think of it like setting up a mathematical equation – we've placed all the pieces in the right order, and now we just need to solve it. Substituting values is a fundamental skill in mathematics, and it's used in various fields, from science and engineering to finance and economics. It allows us to apply mathematical models to real-world scenarios and make predictions. So, now that we've substituted the values, let's focus on performing the calculations accurately and arriving at the final answer. Remember, each step we take is a step closer to the solution, and by carefully following the order of operations, we can ensure that we get the correct result.
Step 3: Calculate the Result
Now, let's calculate the values. 3^2 is 3 times 3, which equals 9. So, 16 * 3^2 becomes 16 * 9. And 2^4 is 2 times 2 times 2 times 2, which equals 16. This is where our arithmetic skills come into play. We're essentially evaluating the powers and then performing the multiplication and division operations. This step is crucial because it's where we actually get the numerical value of the expression. Think of it like cooking a dish – we've prepared the ingredients, and now we're putting them together to create the final product. Calculating the result requires accuracy and attention to detail, so let's take our time and make sure we perform the operations correctly. Understanding the order of operations (PEMDAS/BODMAS) is essential here to ensure that we evaluate the expression in the correct sequence. So, let's carefully perform the calculations and see what the final answer is. Remember, each step we take brings us closer to the solution, and by mastering these arithmetic skills, we're building a solid foundation for more advanced mathematical concepts.
Our expression is now (16 * 9) / 16. Now, 16 times 9 is 144. So, we have 144 / 16. Finally, 144 divided by 16 is 9. We've done it! This is the final step where we simplify the numerical expression to obtain the answer. It's like solving a puzzle – we've put all the pieces together, and now we're revealing the final image. Performing the division is the last step in our journey, and it gives us the numerical value of the expression for the given values of x and y. This step requires careful calculation, and it's important to double-check our work to ensure accuracy. We've now successfully evaluated the expression, and we've arrived at the final answer. Let's celebrate our accomplishment and move on to the next challenge!
Final Answer
Therefore, the value of the expression (2^2 x^2 / x y2)2 for x = 3 and y = 2 is 9. Woohoo! You nailed it! We've successfully evaluated the expression step-by-step, simplifying it, substituting the values, and calculating the final result. This problem demonstrates the power of breaking down complex tasks into smaller, more manageable steps. By following the order of operations and applying the rules of exponents and algebra, we were able to arrive at the correct answer. This kind of problem is a great way to reinforce your understanding of fundamental mathematical concepts and build your problem-solving skills. Remember, practice makes perfect, so keep tackling similar problems and you'll become a pro in no time. We've now added another tool to our mathematical toolkit, and we're ready to take on more challenging problems. Let's continue to explore the fascinating world of mathematics and discover the beauty and elegance of numbers and equations.
Evaluate the expression (2^2 x^2 / x y2)2 when x is 3 and y is 2.
Evaluate (2^2 x^2 / x y2)2 for x=3, y=2: Step-by-Step Solution