Simplifying Expressions: A Step-by-Step Guide

Hey math enthusiasts! Let's dive into a cool algebra problem today. We're going to simplify the expression: $\left(-5 x^4\right)^2\left(2 x^3\right)$. Don't worry, it looks more complicated than it actually is. We'll break it down step by step, making sure everyone understands the process. This is a classic example of how to handle exponents and coefficients together, and it's super important for your algebra journey. So, grab your pencils, and let's get started. By the end of this, you'll be simplifying expressions like a pro, and maybe even impressing your friends with your math skills. Ready? Let's go!

Understanding the Basics: Exponents and Coefficients

Alright, before we jump into the main problem, let's quickly recap some essential concepts. This will help us avoid any confusion down the road. First off, let's talk about exponents. An exponent tells us how many times we multiply a number by itself. For example, in the expression $x^2$, the exponent is 2, which means we multiply x by itself twice: $x * x$. Simple, right? Now, let's consider coefficients. A coefficient is the number that multiplies a variable. In the expression $3x$, the coefficient is 3. This means we're multiplying x by 3. Got it? Okay, because these two concepts are key to solving our main problem. Keep these definitions in mind, they are the building blocks of algebraic expressions. Remember, practice makes perfect, so don't hesitate to work through a bunch of example problems on your own after this. Let's make sure we're all on the same page before we get too far into the weeds.

Breaking Down the Expression

Now, let's return to our original problem, $\left(-5 x^4\right)^2\left(2 x^3\right)$. The first thing we want to do is handle the exponent on the first term, $\left(-5 x^4\right)^2$. Notice that the whole term inside the parenthesis, including both the coefficient and the variable with its exponent, is being raised to the power of 2. We have to address this exponent before multiplying by the second term. When you have a term raised to a power, you apply the exponent to both the coefficient and the variable. What we're actually doing here is squaring everything inside the parentheses. So, let's break it down. We square the coefficient: $(-5)^2 = 25$. Then, we square the variable part. When you raise a power to a power, you multiply the exponents. In our case, $(x^4)^2 = x^{(4*2)} = x^8$. So, $\left(-5 x^4\right)^2$ simplifies to $25x^8$. Great job, guys! We're making progress. Take a moment to really understand what we've done here. We've applied the rules of exponents to simplify the first part of the expression. Always remember to follow the order of operations and to distribute the exponent to everything inside the parentheses. This step is critical; if you mess this up, the rest of your solution will be incorrect. You need to be confident and comfortable with these basic rules. Let's make sure we're all clear before we move on. Practice some similar problems to solidify your understanding.

Putting it All Together: Final Simplification

Okay, now that we've simplified $\left(-5 x^4\right)^2$ to $25x^8$, our original expression \left(-5 x^4\right)^2\left(2 x^3 ight) now looks like this: $25x^8 * (2x^3)$. Now we multiply the terms. We multiply the coefficients: $25 * 2 = 50$. Then, we multiply the variable parts. When multiplying variables with exponents, you add the exponents. So, $x^8 * x^3 = x^{(8+3)} = x^{11}$. Combining these results, we get our final answer: $50x^{11}$. And that's it! We've successfully simplified the expression. See, wasn't that fun? We started with a complex-looking expression and, by applying the rules of exponents and coefficients step-by-step, we arrived at a much simpler form. Remember the main rules: when raising a power to a power, multiply the exponents; when multiplying variables with exponents, add the exponents; and always take care to apply the exponent to all terms inside the parentheses. You've now got another tool in your mathematical toolkit. Congratulations! You've just simplified an algebraic expression with confidence. You're well on your way to becoming a math whiz. Keep up the great work, and you'll find that these kinds of problems become easier and easier.

The Correct Answer and Why

So, after all the calculations, we've found our answer. The simplified form of \left(-5 x^4\right)^2\left(2 x^3 ight) is $50x^{11}$. Looking at the multiple-choice options, the correct answer is (C). To recap, we first handled the exponent on the first term. This meant squaring both the coefficient and the variable part. Then, we multiplied the simplified first term by the second term. When we multiplied the variable parts, we added the exponents. This gave us our final answer. It's really all about applying the rules systematically and carefully. Always take your time and double-check your work to avoid making careless mistakes. Make sure that you understand the process and can explain it to someone else. This is a great way to solidify your understanding. When you can teach the concept to someone else, you know that you truly grasp it. Now that you've worked through this problem, you should be able to tackle similar problems with confidence. The key is to break down the problem into smaller steps and apply the rules of exponents and coefficients correctly. Excellent work, everyone!

Additional Tips for Success

Want to become an even better problem-solver? Here are a few extra tips. First, practice regularly. The more you practice, the more comfortable you'll become with these types of problems. Work through various examples, changing up the coefficients and exponents to keep things interesting. Second, always show your work. Writing down each step helps you avoid mistakes and makes it easier to spot any errors if you get the wrong answer. This also helps when you need to review and refresh your knowledge later. Third, understand the concepts. Don't just memorize the rules. Make sure you understand why they work. This will help you in the long run. Finally, don't be afraid to ask for help. If you're struggling with a problem, reach out to your teacher, a tutor, or a study group. Math can be tricky, but there are always resources available to help you succeed. Keep these tips in mind as you continue your math journey, and you'll be well on your way to mastering algebraic expressions and more. Remember that consistent effort and a positive attitude are key. Good luck, and keep up the great work!