Equivalent Expression To (4x³y⁵)(3x⁵y)² Explained
Hey guys! Let's dive into this math problem together and figure out which expression is equivalent to the given one. We're going to break it down step-by-step, so it's super easy to follow along. Math can seem daunting, but trust me, with a bit of patience, we'll nail it! Our mission is to simplify the expression $\left(4 x^3 y^5\right)\left(3 x^5 y\right)^2$ and match it with one of the options provided. So, let’s jump right into it!
Understanding the Basics
Before we tackle the main problem, it's crucial to brush up on some fundamental concepts. We'll be using the rules of exponents extensively, so having a solid grasp of these will make the whole process a breeze. Exponents, at their core, represent repeated multiplication. For instance, $x^3$ means $x * x * x$. When we're dealing with expressions involving exponents, there are a few key rules to keep in mind. The product of powers rule states that when you multiply terms with the same base, you add their exponents: $x^a * x^b = x^a+b}$. For example, $x^2 * x^3 = x^{2+3} = x^5$. Next up is the power of a power rule, which says that when you raise a power to another power, you multiply the exponents$. So, $(x2)3 = x^{2*3} = x^6$. Lastly, the power of a product rule tells us that when you raise a product to a power, you raise each factor to that power: $(xy)^a = x^a y^a$. For example, $(2x)^3 = 2^3 x^3 = 8x^3$. These rules are the bread and butter of simplifying exponential expressions. They allow us to manipulate terms and combine like factors, making complex problems much more manageable. Mastering these rules is not just about solving this particular problem; it’s about building a strong foundation for more advanced math topics. Think of it as equipping yourself with the right tools for any algebraic challenge that comes your way. With these tools in our arsenal, we can confidently approach the given expression and simplify it with precision.
Breaking Down the Expression
Now that we've refreshed our understanding of exponent rules, let's get our hands dirty with the expression at hand: $\left(4 x^3 y^5\right)\left(3 x^5 y\right)^2$. The first thing we're going to do is tackle that squared term, $(3x5y)2$. Remember the power of a product rule? It tells us that we need to apply the exponent to each factor inside the parentheses. So, we square the 3, square the $x^5$, and square the $y$. Let's break it down: $(3x5y)2 = 3^2 * (x5)2 * y^2$. Now, $3^2$ is simply 9. Next, we use the power of a power rule on $(x5)2$, which means we multiply the exponents: $(x5)2 = x^5*2} = x^{10}$. And finally, $y^2$ remains as is. Putting it all together, we havey^2$. Fantastic! We've successfully simplified the squared term. Now, we can rewrite the original expression with this simplified term: $\left(4 x^3 y^5\right)\left(9 x^{10} y^2\right)$. This looks much more manageable, doesn't it? We're one step closer to the final simplified form. The key here was to take it one step at a time, applying the exponent rules systematically. By breaking down the complex term into smaller, easier-to-handle parts, we avoided getting overwhelmed and ensured accuracy. This approach is crucial in any math problem – dissecting it into manageable chunks makes the solution much clearer and less intimidating. So, let's carry this momentum forward as we combine the remaining terms and reach our final answer.
Combining Like Terms
Alright, guys, we've simplified our expression to $\left(4 x^3 y^5\right)\left(9 x^10} y^2\right)$. Now comes the fun part – combining those like terms! Remember, we can only combine terms that have the same base. In this case, we'll be combining the $x$ terms and the $y$ terms separately. First, let’s focus on the coefficients, which are the numbers in front of the variables. We have 4 and 9, so we simply multiply them together$. This is where the product of powers rule comes into play. We add the exponents: $x^3 * x^10} = x^{3+10} = x^{13}$. Great! We've handled the $x$ terms. Next up are the $y$ terms. We have $y^5$ and $y^2$. Again, we use the product of powers rule and add the exponents = y^7$. Awesome! We've taken care of the $y$ terms as well. Now, all that’s left is to put everything together. We have the coefficient 36, the $x$ term $x^13}$, and the $y$ term $y^7$. Combining these, we get our simplified expressiony^7$. And there you have it! We've successfully combined all the like terms and arrived at a clean, simplified expression. This step highlights the importance of paying attention to the details and applying the exponent rules correctly. By systematically multiplying the coefficients and adding the exponents of like variables, we transformed a seemingly complex expression into a much simpler form. This is a testament to the power of breaking down problems into smaller, manageable steps. Let's move on to the final stage where we match our result with the given options.
Matching the Result
Okay, team, we've done the hard work and simplified the expression to $36x{13}y7$. Now, it's time to play the matching game and see which of the given options aligns with our result. Let's take a look at the options:
A. $24 x^{13} y^7$ B. $36 x^{13} y^7$ C. $36 x^{28} y^6$ D. $144 x^{16} y^{12}$
Scanning through the options, we can immediately spot a familiar face. Option B, $36 x^{13} y^7$, is a perfect match for our simplified expression. High five! We nailed it! Options A, C, and D, on the other hand, don't quite fit the bill. Option A has the correct exponents but the coefficient is off. Options C and D have different exponents altogether, so they're definitely not the equivalent expression we're looking for. This step is crucial because it reinforces the importance of accuracy throughout the simplification process. If we had made a mistake in any of the previous steps, our final simplified expression would be different, and we might have incorrectly chosen the wrong option. By carefully applying the exponent rules and combining like terms, we ensured that our result was correct and that we could confidently identify the matching option. This also highlights the value of double-checking your work. A quick review of your steps can often catch any minor errors and prevent you from selecting the wrong answer. So, with a triumphant flourish, we can confidently say that Option B is the winner!
Final Answer
So, there you have it, folks! We've successfully navigated this mathematical journey, starting from a seemingly complex expression and simplifying it step-by-step to arrive at the correct answer. The equivalent expression to $\left(4 x^3 y^5\right)\left(3 x^5 y\right)^2$ is indeed B. $36 x^{13} y^7$. We began by refreshing our understanding of exponent rules, which are the fundamental tools for simplifying such expressions. We then broke down the original expression, tackled the squared term using the power of a product and power of a power rules, and combined like terms by multiplying coefficients and adding exponents. Finally, we matched our simplified expression with the given options, confirming our hard-earned result. This problem serves as a fantastic example of how breaking down complex problems into smaller, more manageable steps can make them much less intimidating. By systematically applying the rules and paying attention to detail, we can conquer any algebraic challenge that comes our way. Remember, math is not about memorizing formulas; it's about understanding the underlying concepts and applying them strategically. So, keep practicing, keep exploring, and keep challenging yourselves. You've got this!