Equivalent Expression Of (5ab)^3 / (30a^-6b^-7)

Hey guys! Today, let's break down a math problem that involves simplifying an algebraic expression. We're going to tackle the expression $\frac{(5 a b)^3}{30 a^{-6} b^{-7}}$ and find out which of the given options is equivalent to it. Don't worry, we'll take it step by step so it's super easy to follow. So, grab your pencils, and let's dive in!

Understanding the Problem

Our main goal here is to simplify the expression $\frac{(5 a b)^3}{30 a^{-6} b^{-7}}$. This involves using the laws of exponents to manipulate the terms and arrive at a simplified form. It might look intimidating at first, but trust me, it's all about breaking it down into smaller, manageable parts. We need to remember a few key exponent rules, such as the power of a product rule, the quotient of powers rule, and how to handle negative exponents. These rules are the bread and butter of simplifying algebraic expressions, and we'll be using them extensively throughout this process. So, keep these rules in mind as we move forward, and you'll see how they help us transform the original expression into something much simpler and easier to understand. Let’s get started and make this expression our math buddy!

Step-by-Step Solution

Let's break this down step-by-step to make it super clear.

Step 1: Apply the Power of a Product Rule

First, we'll focus on the numerator, $(5ab)^3$. The power of a product rule states that $(xy)^n = x^n y^n$. Applying this rule, we get:

$(5 a b)^3 = 5^3 a^3 b^3 = 125 a^3 b^3$

So, we've taken the first step in simplifying our expression. By applying the power of a product rule, we've expanded the numerator and made it easier to work with. This is a crucial step because it allows us to deal with each term individually, which simplifies the overall process. Remember, the key to solving complex problems is often breaking them down into smaller, more manageable parts. Now that we've handled the numerator, let's move on to the next step and see how we can further simplify the expression.

Step 2: Rewrite the Expression

Now we can rewrite the entire expression using the simplified numerator:

$\frac{(5 a b)^3}{30 a^{-6} b^{-7}} = \frac{125 a^3 b^3}{30 a^{-6} b^{-7}}$

This step is all about putting the pieces together. We've taken the simplified form of the numerator and plugged it back into the original expression. This gives us a clearer picture of what we're working with and sets the stage for further simplification. By rewriting the expression, we've made it easier to see the individual components and how they relate to each other. This is a common strategy in problem-solving – taking a step back and looking at the whole picture before diving into the next step. So, with the expression rewritten, we're ready to move on and tackle the next challenge in our simplification journey.

Step 3: Use the Quotient of Powers Rule

Next, we'll simplify the expression by using the quotient of powers rule, which states that $\frac{x^m}{x^n} = x^{m-n}$. We'll apply this rule to both the 'a' and 'b' terms:

$\frac{a^3}{a^{-6}} = a^{3 - (-6)} = a^{3 + 6} = a^9$

$\frac{b^3}{b^{-7}} = b^{3 - (-7)} = b^{3 + 7} = b^{10}$

The quotient of powers rule is a powerful tool for simplifying expressions with exponents. By applying this rule, we've managed to combine the 'a' terms and the 'b' terms, making our expression significantly cleaner. This step demonstrates the elegance of exponent rules – they allow us to manipulate expressions in a systematic way and reduce them to their simplest forms. Notice how the negative exponents in the denominator become positive when we subtract them. This is a common trick that's super useful in simplifying algebraic expressions. Now that we've handled the variables, let's move on to the numerical coefficients and see how we can simplify those as well.

Step 4: Simplify the Coefficients

Now let's simplify the numerical coefficients:

$\frac{125}{30} = \frac{25 imes 5}{6 imes 5} = \frac{25}{6}$

Simplifying the coefficients is a straightforward step that helps us clean up the expression further. By dividing both the numerator and the denominator by their greatest common factor (which is 5 in this case), we've reduced the fraction to its simplest form. This is an important step in any simplification process, as it ensures that our final answer is in the most concise form possible. Remember, fractions are often easier to work with when they are in their simplest form. So, by simplifying the coefficients, we're making our lives easier and paving the way for the final step in solving the problem. Let's keep going and bring it all together!

Step 5: Combine Simplified Terms

Finally, let's put everything together. We combine the simplified coefficients and the simplified variable terms:

$\frac{125 a^3 b^3}{30 a^{-6} b^{-7}} = \frac{25}{6} a^9 b^{10}$

So, the equivalent expression is $\frac{25 a^9 b^{10}}{6}$.

Combining the simplified terms is the final flourish in our simplification journey. We've taken all the individual components that we've worked on – the simplified coefficients, the 'a' terms, and the 'b' terms – and brought them together to form our final answer. This step is like the grand finale of a fireworks show, where all the individual elements come together to create a spectacular display. By combining the terms, we've transformed the original complex expression into a much simpler and more manageable form. This is the ultimate goal of simplification, and it's incredibly satisfying to see it all come together. So, let's celebrate our success and move on to the next section where we'll discuss why this answer is correct and how it aligns with the given options.

Identifying the Correct Option

Looking back at the options, we can see that:

A. $\frac{a^7 b^{10}}{6}$ is incorrect. B. $\frac{125 a^{18} b^{21}}{30}$ is incorrect. C. $\frac{25 a^3 b^4}{6}$ is incorrect. D. $\frac{25 a^9 b^{10}}{6}$ is the correct answer.

So, option D is the winner! We've successfully simplified the expression and matched it to the correct option. But let's not stop here. It's always a good idea to reflect on the process and understand why the other options are incorrect. This helps us solidify our understanding and avoid similar mistakes in the future. Remember, learning from mistakes is a crucial part of mastering any skill, and math is no exception. So, let's take a closer look at why options A, B, and C don't fit the bill, and reinforce our understanding of the correct solution.

Why Other Options Are Incorrect

Let's quickly go over why the other options are not equivalent to the given expression. Understanding why incorrect options are wrong is just as important as knowing why the correct option is right. It helps reinforce the concepts and prevents you from making similar mistakes in the future. Think of it as building a strong foundation for your math skills – the more you understand, the more confident you'll become in tackling complex problems.

• Option A $\frac{a^7 b^{10}}{6}$: This option incorrectly calculates the exponent of 'a'. The correct exponent should be 9, not 7.
• Option B $\frac{125 a^{18} b^{21}}{30}$: This option seems to have incorrectly applied the power of a product rule and made mistakes in the exponents of both 'a' and 'b'.
• Option C $\frac{25 a^3 b^4}{6}$: This option incorrectly calculates the exponents of both 'a' and 'b'. The correct exponents should be 9 and 10, respectively.

By pinpointing the errors in each incorrect option, we can see exactly where the potential pitfalls lie in simplifying expressions. This kind of analysis is invaluable for improving our problem-solving skills and ensuring that we arrive at the correct answer every time. So, remember to always double-check your work and think critically about each step you take. With practice and careful attention to detail, you'll become a pro at simplifying algebraic expressions!

Conclusion

In conclusion, guys, we've successfully simplified the expression $\frac{(5 a b)^3}{30 a^{-6} b^{-7}}$ to $\frac{25 a^9 b^{10}}{6}$. We did this by applying the power of a product rule, the quotient of powers rule, and simplifying coefficients. Remember, the key to solving these problems is to break them down into smaller steps and apply the exponent rules correctly. Keep practicing, and you'll become a pro at simplifying expressions in no time!

Simplifying algebraic expressions might seem like a daunting task at first, but with the right approach and a solid understanding of the rules, it becomes much more manageable. We've walked through each step of the process, from expanding the numerator to combining the simplified terms, and hopefully, you've gained a clear understanding of how to tackle similar problems. Remember, math is like a muscle – the more you exercise it, the stronger it becomes. So, keep practicing, keep exploring, and most importantly, keep having fun with it! You've got this! And who knows, maybe the next time you encounter a complex expression, you'll be the one explaining it to others. Keep up the great work, guys!