Problem 16 - Fundamentals of Mathematics

Question

Find each value. (-14)\(\cdot(-3)\)

Step-by-Step Solution

Answer

The product of (-14) and (-3) is 42.

Step 1: Identify the Signs of Each Number

Look at the numbers involved in the multiplication: -14 and -3. Both numbers are negative.

Step 2: Apply the Rule of Signs

Recall the multiplication rule of signs: when two negative numbers are multiplied, the result is a positive number. Therefore, the product of -14 and -3 will be positive.

Step 3: Multiply the Absolute Values

Ignore the signs and multiply the absolute values of the two numbers. Calculate 14 multiplied by 3. \[ 14 \times 3 = 42 \]

Step 4: Apply the Resulting Sign

Based on Step 2, since we are multiplying two negative numbers, the sign of the result is positive. Therefore, the final answer is positive 42.

Key Concepts

Understanding Negative Numbers

Negative numbers are numbers that are less than zero, and they often represent opposite values or directions in real-world situations. When you see a number with a minus sign in front of it, such as \(-14\) or \(-3\), that indicates the number is negative.

• Negative numbers are used in various contexts, like measuring temperature below zero, describing debts, or indicating direction.
• In our example, both \(-14\) and \(-3\) are negative numbers.
• Handling them properly in mathematics is crucial to getting correct results.

Recognizing that both numbers in the expression \((-14) \cdot (-3)\) are negative is the first step in solving the problem.

Rule of Signs in Multiplication

The rule of signs is a mathematical principle applied during multiplication and division. It determines the sign of the result based on the signs of the numbers involved.

• When you multiply two numbers with the same sign, either both positive or both negative, the result is always positive.
• Conversely, when you multiply two numbers with different signs, one positive and one negative, the result becomes negative.

With this rule, we can understand why \((-14) \cdot (-3)\) results in a positive number. Despite both numbers being negative, their product is positive. Knowing this rule helps simplify the multiplication process by focusing on the absolute values of the numbers.

Absolute Value Multiplication

Absolute value refers to the magnitude or "size" of a number, disregarding its sign. For example, the absolute value of \(-14\) and \(14\) is the same, which is \(14\).

• To handle multiplication involving negative numbers, calculate using their absolute values.
• The absolute value allows you to avoid complicating the calculation with signs until determining the result’s sign with the rule of signs.

In the given exercise \((-14) \cdot (-3)\), we calculate the multiplication of absolute values: \[ 14 \times 3 = 42 \] This step simplifies finding the product, which is then adjusted with the rule of signs to ensure the correct sign in the final answer, which is \(+42\). Remember, working with absolute values is a helpful tool to reduce errors when working with negative numbers.