Problem 38 - Algebra for College Students

Question

Perform the following operations with real numbers. $$(-8.5)(-3.3)$$

Step-by-Step Solution

Answer

The result of \((-8.5)(-3.3)\) is \(28.05\).

Step 1: Identify the operation

The operation involved is the multiplication of two negative real numbers: \(-8.5\) and \(-3.3\).

Step 2: Apply rules for multiplying negative numbers

Recall that the product of two negative numbers is a positive number.

Step 3: Calculate the product

Multiply the absolute values of the numbers: \[8.5 imes 3.3\].

Step 4: Perform multiplication

Calculate \(8.5 imes 3.3\). First, multiply \(8.5 imes 3 = 25.5\), then multiply \(8.5 imes 0.3 = 2.55\). Finally, add the results: \(25.5 + 2.55 = 28.05\).

Step 5: State the final result

Since the result of multiplying two negative numbers is positive, the final answer is \(28.05\).

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero, represented with a minus sign (-). They are found to the left of zero on the number line.
Negative numbers can represent quantities like debt, temperatures below freezing, or depths below sea level.
Understanding negative numbers is essential because they enable us to solve real-world problems that involve low or reverse quantities.

• When you see a negative number, it's important to recognize that it reflects a direction opposite to positives.
• In calculations, especially in multiplication, handling them properly determines whether the result is negative or positive.

Multiplication Rules

Making sense of multiplication rules is key to working with different types of numbers, including negative numbers. There are a few basic rules for multiplying positive and negative numbers:

• When two positive numbers are multiplied, the result is positive.
• When a positive and a negative number are multiplied, the result is negative.
• When two negative numbers are multiplied, the result is positive.

These rules are consistent and apply to all numbers, whether whole numbers or decimals, like in our exercise with (-8.5) and (-3.3). This understanding allows for consistency in mathematical operations.

Absolute Value

The absolute value of a number is its distance from zero on the number line, without considering the direction. In simpler terms, it’s always a non-negative number.
For example, the absolute value of -3 and 3 is 3. It is symbolized with vertical bars, like this: |x|.

• The absolute value helps us handle negative numbers in multiplication by allowing a focus on magnitude alone without signs interfering during the calculation.
• In our provided exercise, you first calculate |8.5| and |3.3| and multiply those values just like positive numbers.

This is critical in obtaining the product's size before applying any rules regarding signs.

Positive Product

The product of two negative numbers creates an interesting scenario.
Even though each of the multiplied numbers is negative, they combine to make a positive product.
This can seem a bit counterintuitive, but it's a fundamental rule in arithmetic.
When you multiply (-8.5) and (-3.3), according to the multiplication rules, the two negatives "cancel out" to ensure the answer is positive.

• The positive product rule is crucial in predicting the outcome of calculations involving negative numbers.
• This rule simplifies tackling various problems across mathematics, ensuring your final step isn't about guessing the sign but focusing instead on understanding the operation.

In summary, having a clear grasp of this positive outcome provides confidence in handling negative multiplications.