Multiplying Integers Step-by-Step Solutions For 6 X (-14), -50 X (-10) X 60, And -16 X (-9) X (-8) X (-7)

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In the realm of mathematics, the multiplication of integers is a fundamental concept that serves as a building block for more advanced topics. Understanding how to multiply positive and negative numbers is crucial for success in algebra, calculus, and beyond. This article delves into the intricacies of integer multiplication, providing a comprehensive guide to solving problems like 6 x (-14), -50 x (-10) x 60, and -16 x (-9) x (-8) x (-7). We will explore the rules governing the multiplication of integers, break down each problem step-by-step, and offer insights into the underlying principles.

Understanding the Rules of Integer Multiplication

The foundation of integer multiplication lies in understanding the rules that govern the signs of the products. These rules are simple yet essential:

  • Positive x Positive = Positive: When two positive integers are multiplied, the result is always a positive integer. For example, 3 x 4 = 12.
  • Negative x Negative = Positive: Multiplying two negative integers also yields a positive result. This is a key concept to grasp. For instance, -2 x -5 = 10.
  • Positive x Negative = Negative: When a positive integer is multiplied by a negative integer, the product is negative. For example, 6 x -7 = -42.
  • Negative x Positive = Negative: Similarly, multiplying a negative integer by a positive integer results in a negative product. For instance, -8 x 9 = -72.

These rules are the bedrock of integer multiplication. By mastering them, you can confidently tackle a wide range of problems.

Solving 6 x (-14)

Let's begin with the first problem: 6 x (-14). This problem involves multiplying a positive integer (6) by a negative integer (-14). According to the rules we just discussed, the product will be negative.

To find the product, we first multiply the absolute values of the numbers: 6 x 14. This can be done through manual calculation or using a calculator. The result is 84.

Since we know the product will be negative, we apply the negative sign to the result. Therefore, 6 x (-14) = -84.

Step-by-step breakdown:

  1. Identify the signs: Positive (6) and Negative (-14).
  2. Determine the sign of the product: Positive x Negative = Negative.
  3. Multiply the absolute values: 6 x 14 = 84.
  4. Apply the negative sign: -84.

Thus, the final answer is -84. This may seem straightforward, but it's vital to understand each step to solve more complex problems.

Solving -50 x (-10) x 60

Next, we'll tackle the problem -50 x (-10) x 60. This problem involves the multiplication of three integers, two of which are negative. To solve this, we'll multiply the numbers in pairs, applying the rules of integer multiplication at each step.

First, let's multiply -50 x (-10). According to the rules, Negative x Negative = Positive. The absolute values are 50 and 10, and their product is 50 x 10 = 500. Since the result is positive, we have -50 x (-10) = 500.

Now, we need to multiply this result (500) by the remaining integer, which is 60. So, we have 500 x 60. This is a straightforward multiplication: 500 x 60 = 30000.

Therefore, -50 x (-10) x 60 = 30000.

Step-by-step breakdown:

  1. Multiply the first two integers: -50 x (-10).
  2. Determine the sign: Negative x Negative = Positive.
  3. Multiply the absolute values: 50 x 10 = 500.
  4. Result: 500.
  5. Multiply the result by the third integer: 500 x 60.
  6. Determine the sign: Positive x Positive = Positive.
  7. Multiply the values: 500 x 60 = 30000.
  8. Final result: 30000.

This example demonstrates how to handle the multiplication of multiple integers by breaking it down into smaller steps.

Solving -16 x (-9) x (-8) x (-7)

Now, let's address the most complex problem: -16 x (-9) x (-8) x (-7). This problem involves the multiplication of four negative integers. To solve this, we will again multiply the numbers in pairs, keeping track of the signs at each step.

First, let's multiply -16 x (-9). As we know, Negative x Negative = Positive. The product of their absolute values is 16 x 9 = 144. So, -16 x (-9) = 144.

Next, we multiply -8 x (-7). Again, Negative x Negative = Positive. The product of their absolute values is 8 x 7 = 56. Thus, -8 x (-7) = 56.

Now, we have reduced the problem to 144 x 56. This is a straightforward multiplication, and the result is 144 x 56 = 8064.

Therefore, -16 x (-9) x (-8) x (-7) = 8064.

Step-by-step breakdown:

  1. Multiply the first two integers: -16 x (-9).
  2. Determine the sign: Negative x Negative = Positive.
  3. Multiply the absolute values: 16 x 9 = 144.
  4. Result: 144.
  5. Multiply the next two integers: -8 x (-7).
  6. Determine the sign: Negative x Negative = Positive.
  7. Multiply the absolute values: 8 x 7 = 56.
  8. Result: 56.
  9. Multiply the two results: 144 x 56.
  10. Determine the sign: Positive x Positive = Positive.
  11. Multiply the values: 144 x 56 = 8064.
  12. Final result: 8064.

This example illustrates how to handle the multiplication of multiple integers by systematically reducing the problem to simpler steps. Recognizing patterns in the signs is key to avoiding errors.

Key Strategies for Integer Multiplication

To master integer multiplication, consider these strategies:

  • Memorize the Sign Rules: The rules for multiplying positive and negative integers are fundamental. Commit them to memory.
  • Break Down Complex Problems: When multiplying multiple integers, break the problem down into pairs. This simplifies the process and reduces the chance of errors.
  • Pay Attention to Signs: Always be mindful of the signs. A single sign error can change the entire result.
  • Practice Regularly: The more you practice, the more comfortable you will become with integer multiplication.
  • Use a Calculator: For larger numbers, don't hesitate to use a calculator to ensure accuracy.

Common Mistakes to Avoid

  • Sign Errors: The most common mistake is overlooking a negative sign or misapplying the sign rules. Double-check your signs at each step.
  • Calculation Errors: Ensure accuracy in your calculations. Use a calculator if necessary.
  • Rushing Through the Problem: Take your time and work through each step carefully. Rushing can lead to errors.

Conclusion

Mastering the multiplication of integers is crucial for success in mathematics. By understanding the rules governing the signs of the products and practicing regularly, you can confidently solve problems like 6 x (-14), -50 x (-10) x 60, and -16 x (-9) x (-8) x (-7). Remember to break down complex problems into smaller steps, pay close attention to signs, and double-check your work. With dedication and practice, you can conquer the world of integer multiplication and build a solid foundation for your mathematical journey.

This comprehensive guide has equipped you with the knowledge and strategies to tackle integer multiplication problems effectively. Keep practicing, and you'll find yourself mastering this essential mathematical skill in no time. Whether you're a student learning the basics or someone looking to brush up on their math skills, understanding integer multiplication is a valuable asset.

By following the steps and strategies outlined in this article, you'll be well-prepared to solve a wide range of integer multiplication problems with confidence and accuracy. Embrace the challenge, and watch your mathematical abilities soar!