Problem 22 - Elementary and Intermediate Algebra

Question

Multiply. \(-13 \cdot(-15)\)

Step-by-Step Solution

Answer

195

Step 1: Identify the numbers

The given numbers are \( -13 \) and \( -15 \).

Step 2: Understand the signs

Both numbers are negative. The rule for multiplying two negative numbers is that their product is a positive number.

Step 3: Multiply the absolute values

Multiply the absolute values of the given numbers: \( | -13 | \) and \( | -15 | \). This gives \( 13 \times 15 \).

Step 4: Perform the multiplication

Calculate \( 13 \times 15 = 195 \).

Step 5: Determine the sign of the product

Since both original numbers were negative, their product is positive. Therefore, \( -13 \times -15 = 195 \).

Key Concepts

absolute values

When we talk about absolute values, we refer to the distance of a number from zero on a number line, regardless of its direction. For example, the absolute value of -13 is 13, and the absolute value of -15 is 15. The absolute value is always a non-negative number.

In mathematical notation, the absolute value of a number x is written as \(|x|\). So, \( |-13| = 13 \) and \( |-15| = 15 \).

Understanding absolute values is crucial when dealing with negative numbers, especially in multiplication. Instead of worrying about the signs right away, you can first focus on multiplying the absolute values to get a positive result. This simplifies the process and makes it easier to manage the signs separately.

multiplication rules

Multiplying numbers might seem straightforward, but applying the correct rules is crucial, especially when dealing with positive and negative numbers. Here is a quick guide to multiplication rules:

• Positive \( \times \) Positive = Positive
• Positive \( \times \) Negative = Negative
• Negative \( \times \) Positive = Negative
• Negative \( \times \) Negative = Positive

In the given exercise, we multiplied two negative numbers: \( -13 \times -15 \). According to the rules, when two negative numbers are multiplied, their product is positive. This rule is consistent and helps keep the multiplication of negative numbers simple and clear.

negative numbers

Negative numbers can sometimes be tricky, but understanding them is important for mastering many math concepts. A negative number is a value less than zero, and it is typically denoted with a minus sign (\( - \)).

When dealing with negative numbers in multiplication:

• It helps to first consider their absolute values.
• After multiplying the absolute values, determine the final product's sign using the multiplication rules.

For example, in the exercise \( -13 \times -15 \):
1. We start with the absolute values, which are 13 and 15.
2. We multiply these to get \( 13 \times 15 = 195 \).
3. Since both original numbers are negative, the product is positive, and thus \( -13 \times -15 = 195 \).

Always remember, a negative times a negative equals a positive!