Equivalent Expression To (st)(6)? Math Problem Solved!

Hey guys! Today, we're diving into a cool math problem that might seem a bit tricky at first, but I promise we'll break it down together. Our main question is: Which expression is equivalent to (st)(6)? This involves understanding the associative property of multiplication, which is a fundamental concept in algebra. So, let's get started and figure out the correct answer while making sure we understand why it's the right one.

Understanding the Problem: Which Expression is Equivalent to (st)(6)?

Okay, so our mission is to find an expression that's the same as (st)(6). What does this even mean? Well, in math, when we see letters next to each other like st, it usually means we're multiplying s and t. The parentheses tell us we're doing that multiplication first. Then, we're multiplying the result by 6. But here's the cool part: the associative property of multiplication says we can change the grouping of the numbers (or variables) we're multiplying without changing the final answer. This property is super useful because it lets us rearrange things to make the math easier. Let's dive deeper into how this works and look at some examples. The key concept here is associative property, which states that the way factors are grouped in a multiplication problem does not change the product. For example, (2 * 3) * 4 is the same as 2 * (3 * 4). Both equal 24. Applying this to our problem, we need to identify an option that rearranges the grouping of s, t, and 6 without altering the fundamental multiplication operation. Think of it like rearranging your furniture – the room stays the same, but the view might be a little different. So, with this in mind, let's look at the possible answers and see which one correctly applies this principle. Remember, we're not changing the order of the numbers, just how they're grouped together.

Analyzing the Options

Let's take a look at the options we have and see which one matches our understanding of the associative property.

A. s(t(6))

This option, s(t(6)), looks pretty promising, right? Let's break it down. It says we should first multiply t by 6, and then multiply the result by s. This is exactly what the associative property allows us to do! We're just changing the order in which we perform the multiplications, but the overall product will be the same. To really nail this down, let's imagine some simple numbers. Say s is 2 and t is 3. In our original expression, (st)(6), we'd have (2 * 3)(6), which is 6 * 6 = 36. Now, let's try option A: s(t(6)). That becomes 2(3 * 6), which is 2 * 18 = 36. See? Same answer! This confirms that option A correctly applies the associative property. It's like saying instead of multiplying the first two numbers and then the third, we multiply the last two first, and then multiply by the first number. The beauty of this is that the order doesn't mess with the final result. This is why understanding these properties is so crucial in math – they give us flexibility and help us simplify complex problems.

B. s(x) × t(6)

Okay, let's check out option B. s(x) × t(6). At first glance, this one seems a little off, doesn't it? The big red flag here is the s(x). In our original problem, we just had s, but now we're seeing s(x). This usually means that s is a function of x, which is a whole different ball game. We're not just multiplying s by something; we're evaluating a function. This completely changes the meaning of the expression. Think about it this way: if s were a function, like s(x) = x + 1, then s(x) would change depending on what x is. This is not the simple multiplication and regrouping we're looking for. To further illustrate why this is incorrect, let's go back to our simple numbers. If we let s = 2 and t = 3 like before, we don't even know how to plug those into s(x) because we have an x now. This clearly deviates from our original expression, which was just about multiplying three terms together. The introduction of a function makes option B an entirely different operation, not just a rearrangement. So, option B doesn't fit the bill because it changes the fundamental nature of the problem.

C. s(6) × t(6)

Now, let's dissect option C. s(6) × t(6). This one is also trying to throw us off! Instead of just multiplying s, t, and 6 together, it's suggesting we first evaluate s at 6 and t at 6, and then multiply those results. This is a big change from our original expression (st)(6). The problem with this approach is that we're not simply rearranging the order of multiplication; we're actually substituting values into s and t. This would only be correct if we knew that s and t were functions of 6 in some way, but we have no information to suggest that. Let's use our numbers again to see why this doesn't work. If we pretend s = 2 and t = 3, then (st)(6) is (2 * 3)(6) = 36, as we've seen. But s(6) × t(6) would mean something completely different. We'd need to know what happens when we plug 6 into s and t, which we don't. This option is essentially trying to change the problem into something else entirely. It's not just a matter of regrouping; it's changing the values we're working with. So, because it alters the fundamental operation from simple multiplication to evaluation at specific values, option C is incorrect.

D. 6 × s(x) × t(x)

Lastly, let's break down option D. 6 × s(x) × t(x). This one is a bit of a mix of the issues we saw in options B and C. First, we have the s(x) and t(x) again, which means we're dealing with functions of x rather than simple variables s and t. This is already a major red flag because it changes the basic structure of the problem. Instead of multiplying constants, we're now dealing with expressions that depend on the value of x. Second, even if we ignored the function aspect, the presence of x where it doesn't belong is a problem. Our original expression, (st)(6), doesn't involve x at all. So, introducing x here is like adding a whole new ingredient to a recipe that wasn't there before. To make it clear, let's go back to our trusty numbers. If s = 2, t = 3, and we stick with our original expression (st)(6), we get (2 * 3)(6) = 36. But with option D, we're not even sure how to plug in these numbers because of the x. This option not only changes the operation by introducing functions but also brings in an extraneous variable. This makes it clear that option D is not equivalent to our original expression. It's a different kind of problem altogether, so it's definitely not the right answer.

The Correct Answer: A. s(t(6))

Alright, after carefully analyzing all the options, it's clear that A. s(t(6)) is the correct answer. This is because it correctly applies the associative property of multiplication. We're simply changing the grouping of the factors without altering the fundamental multiplication operation. Remember, the associative property lets us regroup numbers (or variables) being multiplied without affecting the final product. Options B, C, and D, on the other hand, introduce changes that go beyond simple regrouping. They either involve functions or substitute values in a way that doesn't maintain the equivalence to the original expression. Option A keeps the integrity of the original problem intact, making it the only choice that fits. So, by understanding and applying the associative property, we've successfully navigated this math problem!

Final Thoughts

So, there you have it! We've not only found the answer to the question