Expressions Equivalent To (-7)(-15)(-5) A Detailed Explanation
In the realm of mathematics, particularly when dealing with integer arithmetic, understanding the equivalence of expressions is crucial. This article delves into the problem of identifying expressions equivalent to (-7)(-15)(-5). We will dissect the original expression, explore various simplification pathways, and meticulously evaluate the given options to determine the correct answer. This exploration will not only provide a solution to the specific question but also enhance your understanding of the associative and commutative properties of multiplication.
Understanding the Original Expression: (-7)(-15)(-5)
To begin, let's break down the original expression: (-7)(-15)(-5). This expression represents the product of three negative integers. A fundamental principle in mathematics dictates that the product of two negative numbers is positive. Thus, we can start by multiplying the first two factors:
(-7) * (-15) = 105
Now, our expression simplifies to:
105 * (-5)
Multiplying these two factors, we get:
105 * (-5) = -525
Therefore, the original expression (-7)(-15)(-5) is equivalent to -525. This result serves as our benchmark for evaluating the options provided. It is important to understand this calculation clearly as it forms the basis for identifying equivalent expressions. This initial calculation showcases the importance of order of operations, even within the context of simple multiplication. By meticulously evaluating the product step-by-step, we minimize the risk of errors and gain a solid foundation for further analysis. The concept of multiplying negative numbers, in particular, is a cornerstone of integer arithmetic, and mastering this skill is vital for solving more complex mathematical problems.
Evaluating Option A: (-7)(-75) and (-1)(525)
Now, let's examine the first option, which presents two expressions: (-7)(-75) and (-1)(525). To determine if these expressions are equivalent to the original expression (-7)(-15)(-5), which we established is equal to -525, we need to evaluate each of them individually.
Evaluating (-7)(-75)
The first expression in option A is (-7)(-75). Multiplying these two negative numbers:
(-7) * (-75) = 525
This result, 525, is the positive counterpart of our target value, -525. Therefore, (-7)(-75) is not equivalent to the original expression.
Evaluating (-1)(525)
The second expression in option A is (-1)(525). Multiplying these two numbers:
(-1) * (525) = -525
This result, -525, matches the value of the original expression. However, for option A to be correct, both expressions must be equivalent to the original. Since (-7)(-75) is not equivalent, option A is incorrect.
This step-by-step evaluation highlights the significance of precision in mathematical calculations. Even a seemingly small difference in sign can lead to a completely different result. In this case, the positive product of (-7) and (-75) immediately disqualifies option A. This meticulous approach ensures that we accurately identify the correct answer by thoroughly analyzing each component of the given options.
Analyzing Option B: (-1)(525) and (35)(-15)
Let's move on to option B, which presents the expressions (-1)(525) and (35)(-15). As before, we need to evaluate each expression separately to determine if they are equivalent to -525.
Evaluating (-1)(525)
We already evaluated (-1)(525) in the analysis of option A and found that:
(-1) * (525) = -525
This expression is indeed equivalent to the original expression.
Evaluating (35)(-15)
Now, let's evaluate the second expression, (35)(-15):
(35) * (-15) = -525
This result, -525, also matches the value of the original expression. Since both expressions in option B are equivalent to -525, option B is a potential correct answer.
This evaluation demonstrates how different combinations of numbers can yield the same product. The expression (35)(-15) achieves the same result as (-1)(525) and the original expression (-7)(-15)(-5). This highlights the flexibility and interconnectedness of mathematical operations. By meticulously performing the calculations, we can confidently identify equivalent expressions and deepen our understanding of numerical relationships.
Scrutinizing Option C: (35)(-15) and (115)(-5)
Next, we will analyze option C, which presents the expressions (35)(-15) and (115)(-5). We already know from our analysis of option B that (35)(-15) is equivalent to -525. Therefore, we only need to evaluate (115)(-5).
Evaluating (115)(-5)
Multiplying these two numbers, we get:
(115) * (-5) = -575
This result, -575, is not equivalent to -525. Consequently, option C is incorrect because one of its expressions does not match the value of the original expression.
This step underscores the importance of meticulously checking each expression in the options. Even if one expression is equivalent, the entire option is incorrect if the other expression deviates from the target value. The calculation (115)(-5) clearly demonstrates this principle, reinforcing the need for careful attention to detail in mathematical problem-solving.
Dissecting Option D: (-1)(525) and (115)(-5)
Finally, let's examine option D, which presents the expressions (-1)(525) and (115)(-5). We have already established that (-1)(525) is equivalent to -525 and that (115)(-5) is equal to -575.
Reviewing Previous Evaluations
From our previous analyses, we know:
- (-1)(525) = -525
- (115)(-5) = -575
Since (115)(-5) is not equivalent to -525, option D is incorrect. This option serves as another example of why both expressions in an option must be equivalent to the original expression for the option to be considered correct.
This final evaluation reinforces the importance of consistency and accuracy in mathematical problem-solving. By drawing upon our previous calculations, we can efficiently eliminate option D and solidify our understanding of the problem. The discrepancy in the value of (115)(-5) definitively rules out this option, highlighting the need for meticulous verification in mathematical analysis.
Conclusion: The Equivalent Expressions
After meticulously evaluating all the options, we have determined that option B is the correct answer. Option B presents the expressions (-1)(525) and (35)(-15), both of which are equivalent to the original expression (-7)(-15)(-5), which equals -525.
Therefore, the correct answer is:
B. (-1)(525) and (35)(-15)
This problem exemplifies the importance of understanding the properties of multiplication and the significance of careful calculation. By breaking down the problem into smaller steps and systematically evaluating each option, we can confidently arrive at the correct solution. The ability to identify equivalent expressions is a fundamental skill in mathematics, and this exercise provides valuable practice in applying this skill.
This detailed analysis demonstrates the process of identifying equivalent expressions in mathematics. By systematically evaluating each option and comparing the results to the original expression, we can confidently determine the correct answer. The principles and techniques used in this analysis can be applied to a wide range of mathematical problems, reinforcing the importance of precision, attention to detail, and a thorough understanding of fundamental concepts.
