Problem 45 - Algebra 2

Question

Find the value of each expression. $$ 6(-3) $$

Step-by-Step Solution

Answer

The value of the expression is -18.

Step 1: Identify the Operation

The expression given is \( 6(-3) \), which indicates multiplication between 6 and -3. Multiplication is the operation we'll perform here.

Step 2: Understand the Sign Rule

When multiplying two numbers with different signs, the result is negative. Since 6 is positive and -3 is negative, the product will be negative.

Step 3: Multiply the Absolute Values

Ignore the signs for now and multiply the absolute values of 6 and 3. The product of 6 and 3 is 18.

Step 4: Apply the Sign to the Product

We previously determined that the result will be negative due to the sign rule. Thus, the result of the multiplication \( 6(-3) \) is \(-18\).

Key Concepts

Sign Rules in Multiplication

When multiplying integers, understanding sign rules is crucial. These rules help determine whether the product of two numbers is positive or negative. Here are the basic sign rules in multiplication of integers:

• If you multiply two positive numbers, the result is positive. For example, \( 3 \times 4 = 12 \).
• If you multiply two negative numbers, the product is also positive. As an example, \( (-3) \times (-4) = 12 \).
• If you multiply a positive number with a negative number, the outcome is always negative. For instance, \( 3 \times (-4) = -12 \) and \( (-3) \times 4 = -12 \). This reflects what happens in our exercise \( 6 \times (-3) = -18 \).

Understanding and applying these rules will help you quickly determine the sign of your product when multiplying integers.

Absolute Value

The concept of absolute value is key to simplifying the multiplication of integers. Absolute value refers to the distance of a number from zero on a number line, regardless of direction. It is always a non-negative value. Here’s why it’s important in multiplication:

• When you multiply numbers, you often consider their absolute values first and apply sign rules later. This makes calculations simpler.
• The absolute value of a number is denoted by two vertical bars. For instance, the absolute value of \(-3\) is \( | -3 | = 3 \) and for \(6\), it is \( | 6 | = 6 \).
• In our exercise, you multiply the absolute values \( 6 \times 3 = 18 \) before considering the sign rule, which ultimately determines the negative sign of the final product.

Absolute values help streamline calculations, leading to an easier understanding of the multiplication of integers.

Negative Numbers

Negative numbers can be tricky, especially in multiplication. Recognizing and working with them involves understanding their role on the number line and in arithmetic operations.

• Negative numbers are below zero on the number line, marking the opposite direction of positive numbers. For instance, while \(3\) is three units to the right of zero, \(-3\) is three units to the left.
• When multiplied, negative numbers follow specific rules that can initially seem counterintuitive, like two negatives producing a positive product.
• In multiplication, every negative number signifies an operation that effectively reverses the direction on the number line. For instance, multiplying a positive number by \(-3\) indicates moving three units in the opposite direction of the initial positive movement.

Understanding negative numbers and their behavior in multiplication allows for mastering more complex arithmetic operations involving integers.