Equivalent Expression To 3(7t): Simple Guide

Let's break down this math problem step by step. Our goal is to find an expression that is the same as $3(7t)$. This involves understanding the basic principles of algebra, specifically the associative property of multiplication. We'll explore how this property allows us to rearrange and simplify expressions to find the correct answer. By the end of this explanation, you'll not only know the solution but also understand the underlying mathematical principles that make it work. This knowledge will be invaluable as you tackle more complex algebraic problems in the future. Remember, mathematics is like building blocks; mastering the fundamentals is key to unlocking more advanced concepts. So, let's dive in and make sure you have a solid grasp of this concept. We will explore each option in detail, explaining why certain options are incorrect and highlighting why the correct answer is indeed the solution. This approach will ensure that you have a comprehensive understanding of how to solve similar problems in the future. Understanding the logic behind each step is much more important than simply memorizing the answer.

Understanding the Problem

The problem asks us to find an expression equivalent to $3(7t)$. This means we need to simplify the given expression and see which of the provided options matches our simplified version. The expression $3(7t)$ means 3 times 7 times $t$. In algebra, when we have a number next to a variable (like $7t$), it means the number is multiplied by the variable. So, the expression represents a multiplication operation. Our task is to perform this multiplication and see which of the given options results in the same value. This involves using the associative property of multiplication, which allows us to change the grouping of factors without changing the product. This property is fundamental to simplifying algebraic expressions and is a crucial concept in mathematics. By understanding this property, you can confidently tackle a wide range of problems involving multiplication and simplification. Let's move forward and explore how we can use this property to solve our specific problem. This will give you a clear understanding of the steps involved and help you avoid common mistakes.

Method

To solve this, we use the associative property of multiplication, which states that $a(bc) = (ab)c$. In our case, $a = 3$, $b = 7$, and $c = t$. So, we can rewrite $3(7t)$ as $(3 e 7)t$. Now, we simply multiply 3 by 7, which equals 21. Therefore, the expression simplifies to $21t$. Now we can assess each answer choice. We must select the answer choice that is equivalent to $21t$. Each incorrect answer choice provides an opportunity to review the rules of algebra. Let's consider what errors might lead someone to choose one of these incorrect options, reinforcing your understanding and minimizing the chance of making similar mistakes in the future. By understanding common mistakes, you can develop strategies to avoid them and approach problems with greater confidence. So, let's move on to evaluating the answer choices and see which one matches our simplified expression.

Evaluating the Options

Let's evaluate each option to determine which one is equivalent to $3(7t)$:

A. $7t + 3$: This option represents addition, not multiplication. It means we are adding 3 to $7t$. This is not the same as multiplying 3 by $7t$. Therefore, this option is incorrect. It's important to distinguish between addition and multiplication in algebraic expressions. Mixing up these operations is a common mistake, so always double-check what the problem is asking you to do. This option is a distractor, designed to catch those who may not fully understand the order of operations or the difference between addition and multiplication. Recognizing this helps you avoid such traps.

B. $21t$: This option represents multiplication and is the result we obtained when simplifying $3(7t)$. Since $3 e 7 = 21$, then $3(7t) = 21t$. Therefore, this option is correct. This option directly matches the result we derived through simplification, confirming that our understanding of the associative property and multiplication is accurate. Choosing this option demonstrates a clear grasp of the concepts involved and the ability to apply them correctly.

C. $t + 21$: This option also represents addition, not multiplication. It means we are adding 21 to $t$. This is not the same as multiplying 3 by $7t$. Therefore, this option is incorrect. Similar to option A, this choice highlights the importance of correctly identifying the operation being performed. Addition and multiplication are distinct operations with different rules and outcomes. This distinction is fundamental to algebraic problem-solving.

D. $t + 10$: This option represents addition and doesn't seem to have any direct relationship to the original expression $3(7t)$. There's no clear way to arrive at $t + 10$ from the given expression through valid algebraic manipulations. Therefore, this option is incorrect. This option serves as a distractor, potentially misleading those who are unsure of the correct process. Recognizing that this option is unrelated to the original expression reinforces your ability to critically evaluate answer choices and eliminate those that are clearly incorrect.

Answer

The correct answer is B. $21t$. This is because $3(7t) = (3 e 7)t = 21t$.

Conclusion

In summary, to find an equivalent expression to $3(7t)$, we use the associative property of multiplication to simplify the expression. By multiplying 3 and 7, we get 21, resulting in the expression $21t$. Therefore, the correct answer is B. $21t$. Understanding the associative property and being able to apply it correctly is essential for simplifying algebraic expressions and solving mathematical problems. Remember to always double-check your work and ensure that you are performing the correct operations. With practice, you'll become more confident in your ability to tackle similar problems. Mathematics is about understanding the underlying principles and applying them consistently. Keep practicing, and you'll continue to improve your skills!

For more in-depth information on algebraic expressions and the associative property, visit a trusted math resource like Khan Academy Algebra.