Problem 260 - Intermediate Algebra

Question

In the following exercises, multiply. $$ (-9.16)(-68.34) $$

Step-by-Step Solution

Answer

626.7744

Step 1 - Understand the Signs

When multiplying two negative numbers, the result is positive. This is because a negative times a negative yields a positive.

Step 2 - Multiply the Absolute Values

Ignore the signs and multiply the absolute values of the numbers. So, multiply 9.16 and 68.34.

Step 3 - Perform the Multiplication

Calculate the product of 9.16 and 68.34 using multiplication: \( 9.16 \times 68.34 = 626.7744 \).

Step 4 - Determine the Final Sign

Since both original numbers were negative, the final product is positive. Thus, the answer is positive 626.7744.

Key Concepts

Absolute Values

When learning about multiplication, understanding absolute values is crucial. The absolute value of a number is simply its distance from zero on the number line, regardless of direction. For example, the absolute value of both -9.16 and 9.16 is 9.16.
Absolute values are always positive or zero, and they make working with numbers, especially negative ones, much easier.
By ignoring the signs temporarily, we can focus on the multiplication process itself and handle the signs separately afterwards.

Multiplication Steps

Let's break down the steps to efficiently multiply two numbers, especially when dealing with negatives:

• Step 1: Understand the Signs - When you multiply two negative numbers, the product is positive. Always remember, a negative times a negative equals a positive.
• Step 2: Multiply the Absolute Values - Ignore the signs temporarily and focus solely on the absolute values of the numbers you are multiplying.

In our exercise, we're multiplying -9.16 by -68.34. Start by ignoring the signs and just multiply 9.16 and 68.34:
\[ 9.16 \times 68.34 = 626.7744 \]
Now, we can address the signs in the next step.

Positive Product of Negatives

After multiplying the absolute values, it's time to determine the sign of the result. This step is crucial for accurate answers.
In our example, both numbers we started with were negative. The rule is straightforward: multiplying two negative numbers results in a positive product.
So, the final product of \( (-9.16) \times (-68.34) = 626.7744 \) is positive.
This rule applies consistently:

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

By understanding and applying these rules, you can confidently tackle multiplication problems involving negative numbers.