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Chapter 1: Problem 80

Discuss the similarities and differences between \(6(-3)-2\) and \(6-3(-2)\)

Short Answer

Expert verified
The first expression results in -20, and the second results in 12. They differ in the order of operations.

Step by step solution

01

Evaluate the first expression

For the first expression, calculate the value of \(6(-3)-2\). Multiply 6 by -3 to get -18. Then subtract 2 from -18. The result is \(-18-2=-20\).
02

Evaluate the second expression

For the second expression, calculate the value of \(6-3(-2)\). First multiply -3 by -2 to get 6. Then subtract this result from 6. This gives \(6-(-6)=6+6=12\).
03

Compare the results

The first expression evaluates to -20, while the second evaluates to 12. Clearly, the results are different.
04

Similarities in the expressions

Both expressions involve the numbers 6, -3, and -2, and both use multiplication and subtraction operations.
05

Differences in the expressions

The primary difference lies in the order and combination of operations. The first expression multiplies 6 by -3 first, then subtracts 2. The second expression subtracts the result of -3 times -2 from 6. This difference in order of operations leads to different results.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multiplication
Multiplication is a fundamental arithmetic operation that combines numbers in a repeated addition format. When you multiply two numbers, you are essentially adding one of those numbers to itself multiple times. For example, multiplying 6 by -3, as in the first expression, means you are adding -3 six times:
  • -3 + -3 + -3 + -3 + -3 + -3 = -18
Conversely, multiplying -3 and -2, as seen in the second expression, can also be understood through repeated addition:
  • -2 + -2 = -4
  • -2 + -4 = -6
Therefore, -3 multiplied by -2 results in 6. Multiplication involving negative numbers flips the product's sign:
  • Positive times Positive = Positive: 6 x 3 = 18
  • Positive times Negative = Negative: 6 x -3 = -18
  • Negative times Negative = Positive: -3 x -2 = 6
Subtraction
Subtraction is an arithmetic operation where you find the difference between two numbers. Subtraction is almost like finding out how much one number exceeds another. For example, in the first expression:
  • 18 - 2 = -20
Here, we combine the value of -18 and subtract 2 to deepen the negative value. Subtracting positive values reduces or makes a number more negative if the initial number is also negative. In the second expression, subtraction appears differently:
  • 6 - (-6) = 12
Mathematically, subtracting a negative number is the same as adding its positive counterpart, so: 6 - (-6) is the same as 6 + 6 = 12.
  • Result: Subtracting a Negative = Adding a Positive
  • Subtracting a Positive from a Negative ==> Further Negative
Negative Numbers
Negative numbers are values less than zero, represented with a minus sign (-). They are essential in a variety of mathematical contexts, including this problem. Here鈥檚 a quick overview:

  • Adding Negative Numbers:
    When you add negative numbers, you're moving to the left on the number line. Combining -3 and -2 results in a value of:
    • -3 - 2 = -5

  • Multiplying With Negative Numbers:
    Using the same method as earlier:
    • Positive * Negative = Negative
    • Negative * Negative = Positive
      • Subtracting Negative Numbers:

        • -3 - (-2) is the same as adding the absolute of -2 back:
          ==> (-3 + 2 = -1) This reversing principle can eliminate negations making it vital to keep operations in check.
          • Reducing errors:
            Rewriting equations to smaller visual stages enhance accuracy.

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