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Chapter 1: Problem 4

Evaluate each of the following expressions, if possible. $$(-6)(+3)$$

Short Answer

Expert verified
-18

Step by step solution

01

Identify the signs of the numbers

The first number is -6 which is negative, and the second number is +3 which is positive.
02

Recall the rule for multiplying signed numbers

When multiplying two numbers with different signs, the result is negative.
03

Multiply the absolute values

Multiply the absolute values of the numbers: \[ | -6 | = 6 \ | +3 | = 3 \] Thus, \[ 6 \times 3 = 18 \]
04

Apply the sign rule

Since -6 and +3 have different signs, the product is negative. Therefore, \((-6)(+3) = -18\).

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Key Concepts

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

absolute values
Let's start with the concept of absolute values. An absolute value of a number is its distance from 0 on a number line, regardless of its direction. It's always a non-negative number. We denote the absolute value of a number by placing vertical bars around it. For example, \(|-6| = 6\) and \(|+3| = 3\). This is because both -6 and +3 are 6 and 3 units away from 0, respectively.

When multiplying signed numbers, you often need to find and multiply the absolute values first before considering the signs. The product of the absolute values gives you the magnitude of the final answer. So, for our example: \(6 \times 3 = 18\).
negative and positive numbers
Understanding negative and positive numbers is crucial for correct multiplication. Positive numbers are those greater than zero. They often don't have a sign in front of them but it is implied as '+'. For example, +3 is a positive number. Negative numbers are less than zero and always have a '-' sign in front of them. For instance, -6 is a negative number.

When you're given an expression like \((-6)(+3)\), the first number is -6 (negative) and the second number is +3 (positive). The signs play a crucial role in determining the final result after the multiplication.
rules for multiplication
The rules for multiplying signed numbers are pretty straightforward but essential to remember. Here's a quick guide:
  • Positive \( \times \) Positive = Positive: When you multiply two positive numbers, the result is always positive. E.g., \(3 \times 2 = 6\).
  • Negative \( \times \) Negative = Positive: When you multiply two negative numbers, the result is positive. E.g., \((-3) \times (-2) = 6\).
  • Positive \( \times \) Negative = Negative: When you multiply a positive number by a negative number, the result is negative. E.g., \(3 \times (-2) = -6\).
  • Negative \( \times \) Positive = Negative: Similar to the above rule, when you multiply a negative number by a positive number, the result is negative. E.g., \((-3) \times 2 = -6\).
For our original expression \((-6) \times (+3)\), we multiply the absolute values \(6 \times 3 = 18\) and then apply the rule that multiplying a negative number by a positive number results in a negative product. Therefore, \(-18\) is our final answer.

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