Simplifying Expressions A Step-by-Step Guide To $(\frac{-8 R^6 S^3}{r})^3$

Hey guys! Today, we're diving into a math problem that looks a bit intimidating but is actually quite manageable once we break it down. We're going to simplify the expression $(\frac{-8 r^6 s^3}{r})^3$. Don't worry; we'll take it step by step so you can follow along easily. Let's get started!

Understanding the Problem

Before we jump into solving, let's make sure we understand what the problem is asking. We have an expression with variables, exponents, and a fraction, all wrapped up in parentheses with an exponent outside. Our goal is to simplify this expression, meaning we want to rewrite it in a simpler form by applying the rules of exponents and arithmetic. This involves dealing with the negative sign, the coefficients, and the variables with their respective powers. It might seem like a lot, but trust me, it's totally doable!

To start, let’s identify the key components of the expression: the fraction inside the parentheses, the terms in the numerator and the denominator, and the exponent outside the parentheses. We need to address each of these methodically. First, we'll simplify the fraction inside the parentheses by dividing the terms with the same base. Remember, when we divide terms with exponents, we subtract the exponents. This is a crucial step in making the expression more manageable. After simplifying the fraction, we'll raise the entire expression to the power of 3, which means we’ll apply the exponent to each term inside the parentheses. This involves multiplying the exponents and paying close attention to the negative sign and the coefficient. By breaking the problem into these smaller, more digestible parts, we can tackle it with confidence and avoid making common mistakes. So, let’s roll up our sleeves and get into the nitty-gritty of simplifying this expression!

Step-by-Step Solution

Step 1: Simplify Inside the Parentheses

The first thing we're going to do is simplify the fraction inside the parentheses. We have $(\frac{-8 r^6 s^3}{r})$. Notice that we have $r^6$ in the numerator and $r$ in the denominator. Remember, $r$ is the same as $r^1$. When we divide terms with the same base, we subtract the exponents. So, we have:

$$\frac{r^6}{r^1} = r^{6-1} = r^5$$

Now, let's rewrite the expression with this simplification:

$$\frac{-8 r^6 s^3}{r} = -8 r^5 s^3$$

Great! We've taken the first step in simplifying our expression. By tackling the fraction inside the parentheses, we've made the whole thing look a lot less scary. This is a classic example of how breaking down a complex problem into smaller parts can make it much easier to handle. We've dealt with the $r$ terms, and now we have a much cleaner expression to work with. This step is crucial because it sets us up for the next part of the problem, which involves applying the exponent outside the parentheses. By simplifying the inside first, we avoid making mistakes with more complex calculations later on. Remember, math is often about taking things one step at a time, and we're doing a fantastic job so far!

Step 2: Apply the Exponent

Now that we've simplified the inside of the parentheses, we have $-8 r^5 s^3$. We need to apply the exponent of 3 to this entire expression. This means we raise each term inside the parentheses to the power of 3. Remember, when we raise a product to a power, we raise each factor to that power. So, we have:

$$(-8 r^5 s^3)^3 = (-8)^3 \cdot (r^5)^3 \cdot (s^3)^3$$

Let's break this down further. First, we need to calculate $(-8)^3$. This means $-8$ multiplied by itself three times:

$$(-8)^3 = -8 \times -8 \times -8 = -512$$

Next, we need to deal with $(r^5)^3$ and $(s^3)^3$. Remember, when we raise a power to a power, we multiply the exponents. So:

$$(r^5)^3 = r^{5 \times 3} = r^{15}$$

$$(s^3)^3 = s^{3 \times 3} = s^9$$

Now, let's put it all together:

$$(-8)^3 \cdot (r^5)^3 \cdot (s^3)^3 = -512 r^{15} s^9$$

Awesome! We've successfully applied the exponent to each term and simplified the expression. This step is where the real magic happens, as we transform the expression into its final simplified form. By carefully applying the exponent rules, we’ve navigated through the powers and variables, and arrived at a clear and concise result. This part of the problem highlights the importance of knowing your exponent rules inside and out. They're the key to unlocking these kinds of simplifications. We've handled the coefficient, the $r$ term, and the $s$ term, and now we have a complete simplified expression. Give yourself a pat on the back – you’re doing great!

Final Answer

So, the simplified expression is:

$$-512 r^{15} s^9$$

Looking at our options, this matches option B.

Why This Answer is Correct

The correct answer is B. $-512 r^{15} s^9$ because we followed the correct order of operations and applied the exponent rules accurately. We first simplified the fraction inside the parentheses and then applied the exponent outside the parentheses to each term. This involved raising the coefficient to the power, multiplying the exponents of the variables, and keeping track of the negative sign. This step-by-step approach ensures that we don't miss any details and arrive at the correct simplification. It's crucial to understand why each step is taken and how it contributes to the final result. By understanding the process, we can confidently tackle similar problems in the future. So, well done for following along and mastering this simplification! Remember, practice makes perfect, so keep at it, and you'll become a pro at these types of problems in no time.

Common Mistakes to Avoid

When simplifying expressions like this, there are a few common mistakes that students often make. Let's go over them so you can avoid these pitfalls:

1. Forgetting to Apply the Exponent to All Terms:

   • One of the most common mistakes is forgetting to apply the exponent to every term inside the parentheses. For example, in the expression $(-8 r^5 s^3)^3$, you need to apply the exponent 3 to the -8, the $r^5$, and the $s^3$. Some people might forget to apply it to the coefficient (-8), which will lead to an incorrect answer. To avoid this, always double-check that you've distributed the exponent to each factor inside the parentheses. It's a small step, but it makes a big difference in the final result. Remember, each term is part of the product and must be raised to the power. By being meticulous, you can sidestep this common error and ensure your solution is spot-on. So, keep an eye out for this, and you'll be on the right track!

2. Incorrectly Multiplying Exponents:

   • When raising a power to a power, you need to multiply the exponents. For example, $(r^5)^3 = r^{5 \times 3} = r^{15}$. A common mistake is to add the exponents instead of multiplying them. Adding exponents is for when you are multiplying terms with the same base (e.g., $r^5 \times r^3 = r^{5+3} = r^8$). To avoid this mistake, remember the rule: power to a power, multiply the exponents. It's a simple rule, but it's essential for getting the correct answer. Keep this distinction clear in your mind, and you'll be simplifying exponents like a pro. This is one of those fundamental rules that, once mastered, will serve you well in all sorts of mathematical contexts. So, let's make sure we've got it down pat!

3. Miscalculating the Sign:

   • When dealing with negative numbers raised to a power, it's crucial to get the sign right. A negative number raised to an odd power will be negative, while a negative number raised to an even power will be positive. In our case, $(-8)^3 = -8 \times -8 \times -8 = -512$, which is negative because the exponent 3 is odd. If you made a mistake and thought it was positive 512, that would change your final answer. To avoid this, take a moment to double-check the sign. Write out the multiplication if you need to, to make sure you have the correct sign. This is a small detail that can have a big impact, so it's worth paying close attention to. By being mindful of the signs, you'll ensure your calculations are accurate and your solutions are solid!

4. Not Simplifying Inside the Parentheses First:

   • Following the order of operations (PEMDAS/BODMAS), you should simplify the expression inside the parentheses before applying the exponent. In our problem, we simplified $\frac{-8 r^6 s^3}{r}$ to $-8 r^5 s^3$ before raising it to the power of 3. If you try to apply the exponent before simplifying inside the parentheses, you might make the problem more complicated than it needs to be. Simplifying first often makes the subsequent steps much easier and reduces the chances of errors. Think of it as preparing the ground before planting – a little effort upfront can lead to a much better harvest. So, always remember to simplify within the parentheses first, and you'll be setting yourself up for success!

By keeping these common mistakes in mind and double-checking your work, you can increase your accuracy and confidence in simplifying expressions. Remember, math is a skill that improves with practice, so keep at it!

Practice Problems

To really nail down these concepts, let's try a couple of practice problems. These are similar to the one we just solved, so you can apply the same techniques. Working through these will help solidify your understanding and build your confidence.

1. Simplify the expression: $(\frac{-2 x^4 y^2}{x})^2$
2. Simplify the expression: $(\frac{9 a^7 b^4}{3a^2})^3$

Try solving these on your own, and then you can check your answers with a friend or teacher. The key is to break each problem down into steps, just like we did in the example. Remember to simplify inside the parentheses first, then apply the exponent, and watch out for those common mistakes we talked about. With a bit of practice, you'll be simplifying expressions like a pro!

Conclusion

Great job, guys! We've walked through simplifying the expression $(\frac{-8 r^6 s^3}{r})^3$ step by step. Remember, the key is to break down complex problems into smaller, manageable parts. Simplify inside the parentheses first, apply the exponent, and watch out for those common mistakes. With practice, you'll become more confident and accurate in your math skills. Keep up the great work, and I'll see you in the next math adventure!