Which Equation Has The Correct Sign On The Product?O A. (-12)4 = 48B. (-9)(-9) = -81O C. 31(-10) = -310O
When evaluating mathematical equations involving multiplication, understanding the correct signs on the products is crucial. Many students and learners often find themselves confused about how negative numbers behave during multiplication and which equations are accurate representations of these operations. In this article, we will analyze the given equations to determine which one correctly displays the sign on the product, providing clear explanations and fundamental rules of multiplication involving positive and negative numbers.
Understanding the Basics of Multiplication Signs
Before diving into the specific equations, it’s essential to review the basic rules governing multiplication signs:
Signs of the Multiplication of Numbers
- Positive × Positive = Positive
- Positive × Negative = Negative
- Negative × Positive = Negative
- Negative × Negative = Positive
These rules are fundamental in determining whether the product of two numbers will be positive or negative.
Applying the Rules to the Equations
Understanding these rules helps evaluate whether the equations presented are correct in terms of their signs:- The first equation involves multiplying a negative number by 4.
- The second involves multiplying two negative numbers.
- The third involves multiplying a positive number by a negative number.
Analyzing Equation A: (-12)4 = 48
This equation appears to be written as "(-12)4," which likely indicates multiplication, but the notation might be ambiguous. Assuming it means multiplication, the correct expression should be "(-12) × 4."
Step-by-Step Evaluation of Equation A
- Identify the numbers: –12 and 4.
- Apply the sign rules: Negative times positive results in a negative product.
- Calculate the magnitude: 12 × 4 = 48.
- Apply the sign: Negative times positive = Negative, so the product should be –48.
- Compare with the given result: 48.
Conclusion: The product of –12 and 4 is –48, not 48. Therefore, Equation A is incorrect in its sign on the product.
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Analyzing Equation B: (-9)(-9) = -81
This equation involves multiplying two negative numbers.
Step-by-Step Evaluation of Equation B
- Numbers involved: –9 and –9.
- Sign rule: Negative × Negative = Positive.
- Calculate the magnitude: 9 × 9 = 81.
- Apply the sign: Because both numbers are negative, the product is positive 81.
- Compare with the given result: –81.
Conclusion: The correct product of –9 and –9 is +81, not –81. Hence, Equation B incorrectly states the sign of the product.
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Analyzing Equation C: 31(-10) = -310
This equation involves multiplying a positive number by a negative number.
Step-by-Step Evaluation of Equation C
- Numbers involved: 31 and –10.
- Sign rule: Positive × Negative = Negative.
- Calculate the magnitude: 31 × 10 = 310.
- Apply the sign: The product is negative because one number is negative.
- Compare with the given result: –310.
Conclusion: The calculation and sign are correct; 31 multiplied by –10 is indeed –310. Therefore, Equation C correctly displays the sign on the product.
Summary and Final Verdict
Based on the analysis above, the only equation that correctly shows the sign on the product is Equation C:
- Equation A is incorrect because the product of –12 and 4 should be –48, not 48.
- Equation B is incorrect because the product of –9 and –9 should be +81, not –81.
- Equation C is correct because 31 multiplied by –10 results in –310, correctly reflecting the sign.
Why Understanding Sign Rules Is Important
Mastering the rules of multiplying positive and negative numbers is essential for various areas of mathematics, including algebra, calculus, and real-world problem-solving. Misinterpreting signs can lead to significant errors, especially in complex calculations.
Practical Tips for Remembering Sign Rules
- Think of the signs as "positive" and "negative" attitudes; multiplying negatives gives a positive "attitude."
- Use the "sign chart" to visualize outcomes:
| Multiplier 1 | Multiplier 2 | Product Sign |
|---|---|---|
| Positive | Positive | Positive |
| Positive | Negative | Negative |
| Negative | Positive | Negative |
| Negative | Negative | Positive |
The Importance of Correct Sign Usage in Mathematics
Correctly understanding and applying sign rules ensures accurate calculations, which is vital for students, professionals, and anyone working with numbers. Whether solving algebraic equations or performing real-world calculations, recognizing the correct sign on products prevents errors and promotes mathematical literacy.
Final Thoughts
In conclusion, among the given equations, Equation C: 31(-10) = -310 is the only one correctly displaying the sign on the product. The other equations incorrectly represent the signs based on basic multiplication rules involving positive and negative numbers.
Remember: Always recall the fundamental rules:
- Positive × Positive = Positive
- Positive × Negative = Negative
- Negative × Positive = Negative
- Negative × Negative = Positive
Mastering these rules will help you quickly determine the correctness of equations involving signs, ensuring accuracy in your mathematical endeavors.
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