Equivalent Expressions Unveiling The Associative Property In Mathematics

In the realm of mathematics, identifying equivalent expressions is a fundamental skill. It allows us to manipulate equations, simplify problems, and gain a deeper understanding of mathematical relationships. This article delves into the question of which expression is equivalent to $(st)(6)$, providing a comprehensive explanation of the underlying principles and the correct answer. We will explore the associative property of multiplication, a key concept that governs how we group factors in a multiplication problem. Understanding this property is crucial not only for solving this specific problem but also for tackling a wide range of algebraic challenges. So, let's embark on this mathematical journey and unravel the mystery of equivalent expressions.

When faced with the expression $(st)(6)$, our goal is to find another expression that yields the same result, regardless of the values of $s$ and $t$. The options presented offer different ways of arranging and combining these factors. To determine the correct equivalent expression, we must carefully consider the order of operations and the properties that govern multiplication. The associative property is our primary tool in this endeavor. It states that the way we group factors in a multiplication problem does not affect the final product. In other words, $(a imes b) imes c$ is equivalent to $a imes (b imes c)$. This seemingly simple principle is a powerful tool for manipulating expressions and simplifying calculations. By applying the associative property, we can rearrange the factors in $(st)(6)$ and identify the expression that maintains the same mathematical meaning. Let's examine each option provided and see how the associative property helps us pinpoint the correct answer. This exploration will not only solve the problem at hand but also solidify our understanding of a core mathematical concept.

Decoding the Options A Step-by-Step Analysis

To accurately determine which expression is equivalent to $(st)(6)$, let's dissect each option and meticulously analyze its structure. This step-by-step approach will help us eliminate incorrect choices and confidently arrive at the correct answer. We'll focus on how the factors $s$, $t$, and $6$ are grouped and combined in each expression, paying close attention to the implications of the associative property.

A. $s(t(6))$

This option presents a regrouping of the factors in $(st)(6)$. The parentheses indicate that we first multiply $t$ by $6$, and then multiply the result by $s$. This arrangement aligns perfectly with the associative property of multiplication, which, as we discussed earlier, allows us to change the grouping of factors without altering the product. Therefore, $s(t(6))$ is indeed equivalent to $(st)(6)$. To illustrate this further, consider assigning arbitrary values to $s$ and $t$. For example, let $s = 2$ and $t = 3$. Then, $(st)(6) = (2 imes 3)(6) = 6 imes 6 = 36$, and $s(t(6)) = 2(3(6)) = 2(18) = 36$. This numerical example reinforces the equivalence established by the associative property. However, we must still examine the remaining options to ensure we have identified the most accurate answer.

B. $s(x) imes t(6)$

This option introduces a variable $x$ that is not present in the original expression, making it fundamentally different from $(st)(6)$. The presence of $s(x)$ suggests a function notation, where $s$ is a function of $x$. This implies a different mathematical context altogether. Moreover, the expression $t(6)$ indicates that $t$ is also being treated as a function, evaluated at $6$. This is a significant departure from the original expression, where $s$ and $t$ are simply variables being multiplied. Therefore, $s(x) imes t(6)$ cannot be equivalent to $(st)(6)$ because it involves different variables and function evaluations that are not part of the initial expression. The introduction of the variable $x$ and the function notation change the mathematical meaning entirely, making this option an incorrect choice.

C. $s(6) imes t(6)$

In this option, both $s$ and $t$ are treated as functions evaluated at $6$. Similar to option B, this interpretation differs significantly from the original expression $(st)(6)$, where $s$ and $t$ are simply variables being multiplied together. The notation $s(6)$ implies that we are substituting the value $6$ into some function $s$, and similarly for $t(6)$. This means that we are not just multiplying the variables $s$ and $t$; rather, we are evaluating functions at a specific point and then multiplying the results. This distinction is crucial. If $s$ and $t$ were functions, the expression $(st)(6)$ would likely imply a different operation, such as the product of the functions evaluated at a variable or a constant. Since the original expression involves simple multiplication of variables, option C is not equivalent.

D. $6 imes s(x) imes t(x)$

This option combines elements of both options B and C, introducing the variable $x$ and treating $s$ and $t$ as functions of $x$. The expression $s(x)$ and $t(x)$ suggest that we are dealing with functions, and the presence of $x$ further reinforces this idea. As with option B, the introduction of $x$ and the function notation deviate from the original expression $(st)(6)$. Multiplying $6$ by $s(x)$ and $t(x)$ involves evaluating these functions at $x$ and then performing the multiplication. This is a completely different operation than simply multiplying the variables $s$ and $t$ by $6$. Therefore, option D is not equivalent to $(st)(6)$ due to the function notation and the introduction of the variable $x$, which alter the mathematical meaning of the expression.

The Verdict Option A Emerges as the Correct Choice

After a thorough examination of each option, it becomes clear that option A, $s(t(6))$, is the expression equivalent to $(st)(6)$. This equivalence stems directly from the associative property of multiplication. As we've established, this property allows us to regroup factors in a multiplication problem without changing the result. Option A demonstrates this property in action by changing the grouping of $s$, $t$, and $6$ while maintaining the same mathematical relationship. The other options, B, C, and D, introduce function notation and the variable $x$, which significantly alter the meaning of the expressions and make them nonequivalent to the original $(st)(6)$.

Solidifying Understanding Additional Insights and Examples

To further solidify our understanding of equivalent expressions and the associative property, let's consider some additional examples. These examples will help us recognize the associative property in various contexts and apply it confidently to solve similar problems.

Example 1 Numerical Application

Consider the expression $(2 imes 3) imes 4$. According to the associative property, this is equivalent to $2 imes (3 imes 4)$. Let's verify this numerically. First, $(2 imes 3) imes 4 = 6 imes 4 = 24$. Next, $2 imes (3 imes 4) = 2 imes 12 = 24$. As we can see, both expressions yield the same result, demonstrating the associative property with concrete numbers. This simple example reinforces the fundamental concept that the grouping of factors does not affect the product.

Example 2 Algebraic Extension

Let's extend this concept to a more complex algebraic expression. Consider $(xy)z$. Applying the associative property, this is equivalent to $x(yz)$. This equivalence holds true regardless of the values of $x$, $y$, and $z$. The associative property is a powerful tool for simplifying algebraic expressions and rearranging terms to make calculations easier. For instance, if we were to evaluate this expression for specific values, such as $x = 2$, $y = 5$, and $z = -1$, we could choose the grouping that simplifies the calculation. In this case, $x(yz) = 2(5 imes -1) = 2(-5) = -10$ might be slightly easier than $(xy)z = (2 imes 5)(-1) = 10(-1) = -10$, though both methods will yield the same answer.

Example 3 Real-World Application

The associative property also has practical applications in real-world scenarios. Imagine you are calculating the total cost of purchasing several items. Suppose you buy 3 items that cost $5 each, and you need to calculate the total cost after a 10% sales tax. You can think of this as $(3 imes 5) imes 1.10$ (calculating the cost of the items first and then applying the tax) or as $3 imes (5 imes 1.10)$ (calculating the cost of one item with tax and then multiplying by the number of items). Both approaches will give you the same final cost, illustrating the associative property in a practical context.

Conclusion Mastering Equivalent Expressions and the Associative Property

In conclusion, the expression equivalent to $(st)(6)$ is indeed $s(t(6))$, as determined by the associative property of multiplication. This property is a cornerstone of algebra and is crucial for simplifying expressions and solving equations. By understanding how the associative property works, we can confidently manipulate factors and regroup terms without altering the mathematical meaning of an expression. The other options presented, B, C, and D, introduced function notation and a new variable, making them nonequivalent to the original expression. Through careful analysis and the application of the associative property, we have successfully identified the correct answer and deepened our understanding of equivalent expressions. This skill is invaluable for tackling more complex mathematical problems and developing a strong foundation in algebra.

Which expression among the options is equivalent to the expression $(st)(6)$?

Equivalent Expressions Unveiling the Associative Property in Mathematics