Simplify (rs)^3(2s)^-2(4r)^4: Math Guide

Introduction

Hey guys! Today, we're diving into the fascinating world of algebraic expressions. We're going to break down and simplify the expression $(r_s)^3(2 s)^{-2}(4 r)^4$. This might look a bit daunting at first glance, but don't worry! We'll take it step by step, using the fundamental rules of exponents and algebra to make it crystal clear. The main goal here is to simplify complex expressions into their most basic forms, which is super useful in various areas of math and science. So, grab your thinking caps, and let's get started on this mathematical adventure! We'll explore each component of the expression, apply exponent rules, and combine like terms to arrive at the simplified version. Remember, the key to mastering algebra is practice, so let's jump right in and tackle this problem together!

Breaking Down the Expression

Okay, let's start by dissecting the expression $(r_s)^3(2 s)^{-2}(4 r)^4$. The first part we see is $(r_s)^3$. This means that everything inside the parentheses, which is $r_s$, is raised to the power of 3. Remember the rule: $(ab)^n = a^n b^n$. So, we can rewrite $(r_s)^3$ as $r^3 s^3$. This is a crucial first step in simplifying the entire expression. By applying this rule, we've effectively distributed the exponent to both variables within the parentheses. This makes it easier to work with each variable individually later on. Now, let's move on to the next part of the expression and see how we can simplify that as well. Breaking down the expression like this helps us tackle it piece by piece, making the whole process much more manageable and less intimidating. We're just getting started, but we're already making progress!

Next up, we have $(2s)^{-2}$. This part involves a negative exponent, which might seem tricky, but it's actually quite straightforward. A negative exponent means we take the reciprocal of the base and change the sign of the exponent. So, $(2s)^{-2}$ is the same as $1/(2s)^2$. Now, we apply the rule we used before: $(ab)^n = a^n b^n$. This gives us $1/(2^2 s^2)$, which simplifies to $1/(4s^2)$. Understanding how to handle negative exponents is essential in simplifying algebraic expressions, so this is a key step. We've transformed a term with a negative exponent into a more manageable fraction, setting us up for further simplification. Let's keep going; we're on a roll!

Finally, we have $(4r)^4$. This is similar to the first part we looked at. We apply the rule $(ab)^n = a^n b^n$ here as well. So, $(4r)^4$ becomes $4^4 r^4$. Now, we need to calculate $4^4$. This means 4 multiplied by itself four times: $4 * 4 * 4 * 4 = 256$. So, $(4r)^4$ simplifies to $256r^4$. We've now broken down each part of the original expression and simplified them individually. This methodical approach is super helpful for tackling complex problems. We're almost ready to put everything back together and see the final simplified form. Great job so far, guys! We're making excellent progress!

Putting It All Together

Alright, we've simplified each part of the expression individually. Now it's time to bring them all together. We had $(r_s)^3$, which simplified to $r^3 s^3$. Then we had $(2s)^{-2}$, which became $1/(4s^2)$. And finally, we had $(4r)^4$, which simplified to $256r^4$. So, the original expression $(r_s)^3(2 s)^{-2}(4 r)^4$ can now be written as $(r^3 s^3) * (1/(4s^2)) * (256r^4)$. This is where the magic happens! We're going to combine these simplified terms to get our final answer. Remember, the key to success here is to take it one step at a time and keep track of your terms. We're almost at the finish line, so let's keep the momentum going!

Now, let's multiply these terms together. We have $(r^3 s^3) * (1/(4s^2)) * (256r^4)$. First, let's rearrange the terms to group similar variables together: $(r^3 * 256r^4) * (s^3 * (1/(4s^2)))$. This makes it easier to see how we can combine the terms. When multiplying terms with the same base, we add the exponents. So, $r^3 * 256r^4$ becomes $256r^(3+4) = 256r^7$. For the s terms, we have $s^3 * (1/(4s^2))$. This is the same as $s^3 / (4s^2)$. When dividing terms with the same base, we subtract the exponents. So, $s^3 / s^2$ becomes $s^(3-2) = s^1 = s$. Don't forget the 4 in the denominator, so we have $s/4$. We're making great progress in combining these terms and simplifying further! Let's move on to the final simplification step.

Final Simplification

Okay, we've combined most of the terms. We now have $256r^7$ multiplied by $s/4$. So, our expression looks like this: $(256r^7) * (s/4)$. To simplify this further, we can divide 256 by 4. What's 256 divided by 4? It's 64! So, our final simplified expression is $64r^7s$. And there you have it! We've taken a complex-looking expression and simplified it step by step to arrive at a much cleaner and more manageable form. This process highlights the power of breaking down problems and applying the rules of exponents and algebra. You guys did an awesome job following along, and remember, practice makes perfect! The more you work with these types of problems, the more confident you'll become. So, keep up the great work, and let's tackle more mathematical challenges in the future!

Conclusion

So, to wrap things up, we started with the expression $(r_s)^3(2 s)^{-2}(4 r)^4$ and, through careful application of exponent rules and algebraic principles, we simplified it to $64r^7s$. This journey demonstrates the importance of breaking down complex problems into smaller, more manageable parts. Each step, from distributing exponents to combining like terms, played a crucial role in reaching our final answer. The ability to simplify algebraic expressions is a fundamental skill in mathematics and has wide-ranging applications in various fields. I hope this walkthrough has helped you understand the process better and given you the confidence to tackle similar problems on your own. Remember, guys, practice is key! Keep exploring, keep learning, and keep simplifying! You've got this!