Problem 39 - Elementary and Intermediate Algebra

Question

Multiply. $$ 3 \cdot(-7) \cdot(-2) \cdot 6 $$

Step-by-Step Solution

Answer

252

Step 1 - Multiply the First Two Numbers

First, multiply the initial two numbers: 3 and -7. \[ 3 \times (-7) = -21 \]

Step 2 - Multiply the Result by the Third Number

Next, multiply the result from Step 1 (-21) by -2. \[ -21 \times (-2) = 42 \]

Step 3 - Multiply the Result by the Last Number

Finally, multiply the current result (42) by 6. \[ 42 \times 6 = 252 \]

Key Concepts

Integer Multiplication

Multiplication is a fundamental arithmetic operation. It extends to all integer types, including positive and negative numbers. \(Integer\) refers to whole numbers, including zero, positive numbers, and negative numbers.
When multiplying integers, the main rules to remember are:

• A positive number multiplied by a positive number equals a positive number.
• A positive number multiplied by a negative number results in a negative number.
• A negative number multiplied by a negative number equals a positive number.

This exercise includes multiplying both positive and negative integers. You will see how these rules apply.

Negative Numbers

Negative numbers represent values less than zero and have unique properties in multiplication.
Here are some key points to understand:

• A negative number is denoted with a minus sign (e.g., -7).
• When multiplying two negative numbers, the result is positive because the two negatives cancel each other out.
• When multiplying a negative number and a positive number, the result is negative, representing a reversal in direction or value.

In the given exercise, observe how multiplying negative numbers impacts the overall product.

Step-by-Step Solution

Let's break down the exercise to illustrate each step clearly:
Step 1 - Multiply the First Two Numbers
First, multiply the initial two numbers: 3 and -7. \(3 \times (-7) = -21\)
Step 2 - Multiply the Result by the Third Number
Next, multiply the result from Step 1 (-21) by -2. The multiplication of two negatives results in a positive: \(-21 \times (-2) = 42\)
Step 3 - Multiply the Result by the Last Number
Finally, multiply the current result (42) by 6. \(42 \times 6 = 252\)
Following these steps ensures you handle each pair of numbers correctly and arrive at the correct final answer. In this exercise, the final product is 252.