Problem 24 - Elementary and Intermediate Algebra

Question

Multiply. $$ 15 \cdot(-43) $$

Step-by-Step Solution

Answer

-645

Step 1: Understand the Problem

To multiply 15 by -43, recognize you are dealing with the multiplication of a positive number and a negative number.

Step 2: Multiply the Absolute Values

Ignore the negative sign for a moment and multiply the absolute values: 15 and 43. \[ 15 \times 43 = 645 \]

Step 3: Apply the Sign Rule

A positive number multiplied by a negative number results in a negative number. Therefore, \[ 15 \times (-43) = -645 \]

Key Concepts

absolute values

When multiplying numbers, especially those that have different signs, the concept of 'absolute values' can be very helpful. The absolute value of a number is its distance from zero on the number line, without considering its sign. For example, the absolute value of both -43 and 43 is 43 because we ignore the negative sign. You can think of absolute values as the 'pure' value of a number, without any positive or negative label. In math notation, the absolute value of a number \( x \) is written as \( | x | \). So, \( | -43 | = 43 \) and \( | 15 | = 15 \).

multiplication rules

Understanding multiplication rules is essential to solving problems correctly. Here are the basic rules for multiplying numbers:

• Multiply the absolute values of the numbers. This means you ignore the signs (positive or negative) and simply multiply their absolute values.
• Apply the sign rules to determine the final sign of the product. These rules are based on the signs of the numbers involved.

To break it down further, consider our example. First, we multiply the absolute values: \( 15 \times 43 \). Ignoring the negative sign of -43, we get: \( 15 \times 43 = 645 \). Next, we apply the sign rules.

sign rules

The sign rules in multiplication help us figure out whether the final answer will be positive or negative:

• Positive × Positive = Positive
• Positive × Negative = Negative
• Negative × Positive = Negative
• Negative × Negative = Positive

So, keeping these rules in mind: In our exercise, we have one positive number (15) and one negative number (-43). According to the sign rules, multiplying a positive number by a negative number results in a negative number. Therefore, 15 × (-43) = -645. Remember these simple rules when determining the sign of your answer in multiplication problems.