Determine The Result Of (4x²-7y²)² A Step-by-Step Guide

by ADMIN 56 views

Hey guys! Ever stumbled upon a mathematical expression that looks intimidating at first glance? Well, let's tackle one together! We're going to break down the expression (4x²-7y²)² step-by-step and make it super easy to understand. Math can be fun, especially when you approach it with a clear strategy.

Understanding the Problem

Our main goal here is to expand the expression (4x²-7y²)². This means we need to multiply the expression (4x²-7y²) by itself. Think of it like this: (4x²-7y²)² = (4x²-7y²)(4x²-7y²). To get to the result, we'll be using a common algebraic identity that will make our lives much easier. This identity is the square of a binomial, which states: (a - b)² = a² - 2ab + b². This formula is our best friend when we're dealing with expressions in this format.

In our case, we can see that a corresponds to 4x² and b corresponds to 7y². So, all we have to do is plug these values into our formula and simplify. Make sure to pay attention to the signs and exponents. A little mistake can throw off the entire result. We'll take it nice and slow, making sure each step is clear. Understanding this foundational principle is crucial, guys, because it pops up everywhere in algebra and beyond. Whether you're calculating areas, working with polynomials, or even diving into calculus later on, mastering the square of a binomial will seriously level up your math game.

Now, before we jump into the calculation, let’s make sure we're all on the same page about why this formula works. When you multiply (a - b) by itself, you're essentially doing (a - b)(a - b). If you use the distributive property (also known as the FOIL method), you get a times a, a times -b, -b times a, and -b times -b. This gives you a² - ab - ba + b², which simplifies to a² - 2ab + b². Understanding the ‘why’ behind the formula helps you remember it better and apply it more confidently in various situations. So, you're not just memorizing a formula; you're understanding the underlying mathematical principle. This deeper understanding makes problem-solving much more intuitive and less like rote memorization.

Step-by-Step Solution

Okay, let's get our hands dirty and solve this thing! Remember, our expression is (4x²-7y²)², and we're using the formula (a - b)² = a² - 2ab + b². First, we need to identify our a and b. As we discussed, a = 4x² and b = 7y². Now, we'll substitute these values into the formula. This gives us (4x²)² - 2(4x²)(7y²) + (7y²)². It looks a bit messy right now, but trust me, it'll clean up nicely. The next step is to simplify each term individually.

Let's start with the first term: (4x²)². When you square a term with both a coefficient and a variable, you need to square both parts. So, 4² = 16, and (x²)² = x⁴. Remember the rule of exponents: when you raise a power to a power, you multiply the exponents. This means the first term simplifies to 16x⁴. See? Not so scary after all! Now, moving on to the second term: -2(4x²)(7y²). Here, we need to multiply the coefficients together and keep the variables separate. So, -2 * 4 * 7 = -56. And we have x² and y², so we just write them as x²y². This makes the second term -56x²y². We're making great progress, guys!

Finally, let's tackle the last term: (7y²)². Just like with the first term, we square both the coefficient and the variable. 7² = 49, and (y²)² = y⁴. So, the last term simplifies to 49y⁴. Now we have all the pieces of our puzzle. Let's put them together. We found that (4x²)² = 16x⁴, -2(4x²)(7y²) = -56x²y², and (7y²)² = 49y⁴. Combining these, we get our final expanded form: 16x⁴ - 56x²y² + 49y⁴. Woohoo! We did it! This is the expanded form of our original expression. It might seem like a lot of steps, but breaking it down like this makes it totally manageable. And each time you practice, you'll get even faster and more confident. Remember, math is a skill, and like any skill, it gets better with practice.

Common Mistakes to Avoid

Hey, we all make mistakes, especially in math! But knowing the common pitfalls can help us avoid them. One frequent error is messing up the signs. In our problem, the '-2ab' term is negative, and it’s super important to keep that negative sign. Forgetting it will change your final answer. Another common mistake is incorrectly applying the exponent rules. Remember, when you raise a power to a power, you multiply the exponents, not add them. So, (x²)² is x⁴, not x². Getting these exponent rules straight is crucial for many algebra problems.

Another place where students often slip up is by not squaring both the coefficient and the variable. For example, in (4x²)², you need to square both the 4 and the x². It’s easy to forget to square the coefficient, but that will lead to an incorrect result. So, always double-check that you've squared everything that needs to be squared. Also, be careful with the order of operations. Remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction)? Make sure you're following the correct order to avoid errors. In our problem, we dealt with the exponents first and then the multiplication and addition.

Lastly, a super common mistake is trying to distribute the square over a sum or difference. Remember, (a - b)² is not the same as a² - b². You need to use the formula a² - 2ab + b². This is a big one, so make sure you're solid on this concept. Guys, avoiding these common mistakes comes down to practice and careful attention to detail. When you're working through problems, take your time, show your steps, and double-check your work. The more you practice, the more these things will become second nature. And don’t be afraid to make mistakes – they're part of the learning process! The important thing is to learn from them and keep going.

Practice Problems

Alright, guys, time to put our knowledge to the test! Practice makes perfect, so let's try a few similar problems to solidify our understanding. Here are a couple of expressions for you to expand using the same method we just learned: (3x² - 5y²)² and (2a² - 9b²)². Remember, the key is to identify your a and b, plug them into the formula (a - b)² = a² - 2ab + b², and simplify carefully. Take your time and show your work – it'll help you catch any mistakes along the way.

For the first problem, (3x² - 5y²)², think of 3x² as your a and 5y² as your b. Substitute these values into the formula, and then carefully square each term and multiply. Pay special attention to the negative sign in the middle term. For the second problem, (2a² - 9b²)², follow the same steps. Here, a is 2a² and b is 9b². Again, make sure you're squaring both the coefficients and the variables correctly. And don't forget that negative sign! Working through these practice problems is super important. It's one thing to understand the steps when someone else is explaining them, but it's another thing to apply them yourself. Doing these problems will really help you internalize the process and build your confidence.

If you're feeling up for a challenge, try mixing things up a bit. Maybe change the coefficients or the exponents, or even try a problem with three terms instead of two. The more you challenge yourself, the better you'll become. And if you get stuck, don't worry! Go back and review the steps we discussed, or ask for help from a teacher, tutor, or friend. The important thing is to keep practicing and keep learning. With a little effort, you'll be expanding binomial squares like a pro in no time!

Conclusion

So there you have it, guys! We've successfully tackled the expression (4x²-7y²)² and expanded it to 16x⁴ - 56x²y² + 49y⁴. We walked through each step, discussed the common mistakes to avoid, and even did some practice problems. Hopefully, you now feel much more confident about expanding binomial squares. Remember, math isn't about memorizing formulas; it's about understanding the concepts and practicing the techniques.

The key takeaway here is the importance of the formula (a - b)² = a² - 2ab + b². This formula is a powerful tool in algebra, and mastering it will help you in countless situations. Also, remember the common pitfalls, like forgetting the negative sign or messing up the exponent rules. By being aware of these mistakes, you can avoid them and ensure you get the correct answer. And most importantly, don't be afraid to practice! The more you work through problems, the more comfortable you'll become with the process.

Math can be challenging, but it's also incredibly rewarding. When you solve a tough problem, you get a real sense of accomplishment. And the skills you learn in math can be applied to so many other areas of life. So, keep practicing, keep learning, and keep challenging yourself. You've got this, guys! And if you ever get stuck, remember there are plenty of resources available to help you, from textbooks and websites to teachers and tutors. Happy calculating!