Unveiling Equivalent Expressions: A Math Exploration

by Alex Johnson 53 views

Hey there, math enthusiasts! Today, we're diving into the fascinating world of equivalent expressions. We'll be tackling the question: Which expressions are equivalent to (-7)(-15)(-5)? This might seem like a straightforward arithmetic problem, but it's a fantastic opportunity to understand the underlying principles of multiplication and how signs interact. Let's break it down and explore the options.

Decoding the Problem: Multiplying Negative Numbers

Before we jump into the answer choices, let's refresh our memory on the rules of multiplying negative numbers. Remember, when you multiply two negative numbers, the result is positive. For example, (-2) * (-3) = 6. On the other hand, when you multiply a positive number and a negative number, the result is negative. For instance, (2) * (-3) = -6.

Now, in our original expression, we have three numbers: -7, -15, and -5. We can approach this in steps. First, let's multiply (-7) and (-15). A negative times a negative equals a positive, so (-7) * (-15) = 105. Now, we have 105 * (-5). A positive number multiplied by a negative number gives us a negative result. Therefore, 105 * (-5) = -525. So, the original expression (-7)(-15)(-5) simplifies to -525. This core understanding is crucial for correctly identifying equivalent expressions.

Now, let's analyze each answer option, keeping in mind that we're looking for expressions that also equal -525. This requires careful calculation and a solid grasp of how negative signs affect the product of numbers. Each option will test our understanding of these rules, making it a valuable exercise in mastering multiplication.

Analyzing the Answer Choices: Step by Step

Let's meticulously analyze each answer choice to see which expressions are equivalent to (-7)(-15)(-5), which we've already determined equals -525. We'll evaluate each option, paying close attention to the signs and the products.

A. (-7)(-75) and (-1)(525)

  • First, let's evaluate (-7)(-75). A negative times a negative results in a positive. So, (-7) * (-75) = 525. That's a positive number, which isn't what we're looking for (we need -525).
  • Next, let's look at (-1)(525). This is a negative number times a positive number, resulting in a negative number. Thus, (-1) * (525) = -525. Here, we see one expression that matches our target result.
  • This option contains both 525 and -525, so it is incorrect.

B. (-1)(525) and (35)(-15)

  • We already know from option A that (-1)(525) = -525. This matches our target, so this is a great sign.
  • Now, let's evaluate (35)(-15). A positive number multiplied by a negative number results in a negative number. (35) * (-15) = -525. Both expressions in this answer choice equal -525.
  • This option contains -525 and -525, so it is correct.

C. (35)(-15) and (115)(-5)

  • We know from option B that (35)(-15) = -525. This is promising.
  • Let's calculate (115)(-5). A positive number multiplied by a negative number gives a negative result. (115) * (-5) = -575. This doesn't match our target (-525).
  • This option is incorrect.

D. (-1)(525) and (115)(-5)

  • We know from previous calculations that (-1)(525) = -525. That checks out.
  • However, we also know from option C that (115)(-5) = -575. This does not match our target value.
  • Thus, option D is incorrect.

The Correct Answer: Diving Deeper

After thorough analysis, the correct answer is B. (-1)(525) and (35)(-15). Both expressions in this option simplify to -525, which is the same as the original expression (-7)(-15)(-5). This showcases our understanding of multiplying negative and positive numbers and how to evaluate expressions. We have now successfully solved the problem and identified the equivalent expressions.

This exercise highlights the importance of understanding the fundamental rules of arithmetic. It also shows us how multiple expressions can have the same value, a crucial concept in algebra and beyond. This is more than just about finding the right answer; it's about solidifying our understanding of the building blocks of mathematics. Remember, practice is key, and each problem we solve brings us closer to mastering these concepts!

Conclusion: Mastering the Math

Congratulations! We've successfully navigated the world of equivalent expressions and multiplication with negative numbers. We've honed our skills in performing calculations and understanding how signs affect products. This is a valuable skill that is central to many mathematical concepts. Keep practicing, keep exploring, and keep challenging yourself! Math is a journey of discovery, and every problem solved brings us closer to a deeper understanding of the world around us. Keep an eye out for more math puzzles and explorations. This is the start of a great adventure. Keep up the excellent work!

For more in-depth learning about the fundamental principles of mathematics and multiplication, check out the resources at Khan Academy https://www.khanacademy.org/. You'll find extensive lessons and practice exercises to further enhance your understanding and skills.