Evaluating -6(4 2/3) Which Expression To Use

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In the realm of mathematics, evaluating expressions is a fundamental skill. It involves applying the correct order of operations and utilizing various properties to simplify and arrive at a final answer. This article delves into the intricacies of evaluating the expression -6(4 2/3), exploring the different approaches and highlighting the correct methodology. We will dissect the given options, providing a comprehensive analysis of each, and ultimately pinpoint the one that accurately reflects the application of the distributive property. This exploration will not only solidify your understanding of expression evaluation but also enhance your ability to tackle similar mathematical challenges with confidence. The journey begins with a clear understanding of the distributive property and its role in simplifying expressions. We will then proceed to apply this property to the given expression, meticulously breaking down each step to ensure clarity and comprehension. By the end of this article, you will have a firm grasp of the correct method for evaluating expressions of this nature and be well-equipped to tackle more complex mathematical problems.

Understanding the Expression -6(4 2/3)

The core of our discussion lies in the expression -6(4 2/3). This expression represents the multiplication of -6 with the mixed number 4 2/3. To effectively evaluate this, we need to understand the underlying principles of arithmetic operations and the properties of numbers. The mixed number 4 2/3 can be converted into an improper fraction, which will facilitate the multiplication process. Alternatively, we can directly apply the distributive property, which is the focus of this article. The distributive property states that a(b + c) = ab + ac, where a, b, and c are any real numbers. This property allows us to distribute the multiplication over addition, simplifying the expression. In our case, -6 is multiplied by the sum of 4 and 2/3. The correct application of the distributive property will lead us to the correct evaluation of the expression. The other options presented may seem plausible at first glance, but a closer examination will reveal that they either misapply the distributive property or introduce incorrect operations. Therefore, a thorough understanding of the distributive property is crucial for correctly evaluating this expression. Let's now delve into the analysis of the given options to identify the one that accurately represents the application of the distributive property.

Analyzing the Options

To effectively evaluate the expression -6(4 2/3), let's dissect each option meticulously:

  • Option A: (-6)(4) + (-6)(2/3) This option presents a promising approach. It appears to correctly apply the distributive property. The -6 is being multiplied separately with both the whole number part (4) and the fractional part (2/3) of the mixed number. This aligns perfectly with the distributive property, which states that a(b + c) = ab + ac. In this case, a = -6, b = 4, and c = 2/3. Therefore, this option seems to be on the right track. However, we need to verify the arithmetic to ensure that the final result is accurate. The multiplication of -6 with 4 gives -24, and the multiplication of -6 with 2/3 gives -4. Adding these two results, -24 and -4, gives -28. This option deserves further consideration as it aligns with the correct methodology.

  • Option B: (-6)(4) × (-6)(2/3) This option introduces a critical error. Instead of adding the products of -6 with 4 and -6 with 2/3, it multiplies them. This is a clear misapplication of the distributive property. The distributive property involves the addition of the products, not multiplication. Multiplying the products will lead to a completely different result, and it is not the correct way to evaluate the expression. Therefore, this option can be immediately ruled out as incorrect. The error lies in the fundamental misunderstanding of the distributive property, which is the cornerstone of evaluating this expression correctly.

  • Option C: (-6 + 4) + (-6 + 2/3) This option deviates significantly from the correct application of the distributive property. It incorrectly adds -6 with 4 and -6 with 2/3. This operation has no basis in the distributive property or any other valid mathematical principle for evaluating this expression. The distributive property involves multiplication, not addition, between the term outside the parentheses and the terms inside. Therefore, this option is clearly incorrect and demonstrates a lack of understanding of the fundamental principles of expression evaluation.

  • Option D: (-6 + 4) × (-6 + 2/3) Similar to option C, this option also misapplies the operations. It adds -6 with 4 and -6 with 2/3, and then multiplies the results. This is not a valid method for evaluating the given expression. The correct approach, as dictated by the distributive property, involves multiplying -6 with both 4 and 2/3 and then adding the products. This option introduces an extraneous addition operation that is not justified by the mathematical principles governing expression evaluation. Therefore, this option is incorrect.

The Correct Application of the Distributive Property

The key to correctly evaluating the expression -6(4 2/3) lies in the proper application of the distributive property. As we discussed earlier, the distributive property states that a(b + c) = ab + ac. To effectively utilize this property, we must first recognize the components of our expression in terms of the distributive property. In our case, -6 is the 'a', 4 is the 'b', and 2/3 is the 'c'. Therefore, we need to multiply -6 with both 4 and 2/3, and then add the resulting products.

This can be represented as:

-6(4 2/3) = (-6)(4) + (-6)(2/3)

Now, let's perform the multiplications:

  • (-6)(4) = -24
  • (-6)(2/3) = -12/3 = -4

Next, we add the products:

-24 + (-4) = -28

Therefore, the correct evaluation of the expression -6(4 2/3) is -28. This result aligns perfectly with the methodology presented in option A, further solidifying its correctness. The other options, as we have analyzed, introduce incorrect operations or misapply the distributive property, leading to inaccurate results. The meticulous application of the distributive property, as demonstrated here, ensures the accurate evaluation of the expression and reinforces the importance of understanding fundamental mathematical principles.

Conclusion

In conclusion, when evaluating the expression -6(4 2/3), the correct approach is to apply the distributive property accurately. Option A, (-6)(4) + (-6)(2/3), correctly demonstrates this principle by multiplying -6 separately with both the whole number and fractional parts of the mixed number and then adding the products. This method aligns perfectly with the distributive property, which is the cornerstone of evaluating such expressions. The other options presented introduce errors in the application of the distributive property or employ incorrect operations, leading to inaccurate results. Therefore, the key takeaway is the importance of understanding and correctly applying fundamental mathematical principles, such as the distributive property, when evaluating expressions. This understanding will not only enable you to solve similar problems with confidence but also lay a strong foundation for tackling more complex mathematical challenges in the future. The ability to accurately evaluate expressions is a crucial skill in mathematics, and mastering the distributive property is a significant step in that direction.