Problem 35 - Elementary and Intermediate Algebra

Question

Multiply. $$ (-5.3)(2.1) $$

Step-by-Step Solution

Answer

-11.13

Step 1: Understand the Signs

Identify the signs of the numbers being multiplied. Here, we have a negative number \(-5.3\) and a positive number \(2.1\).

Step 2: Multiply the Absolute Values

Ignore the signs and multiply the absolute values of the numbers. Calculate \(5.3 \times 2.1\).

Step 3: Perform the Multiplication

Perform the multiplication: \(5.3 \times 2.1 = 11.13\).

Step 4: Determine the Sign of the Product

Since the original numbers had different signs (one negative and one positive), the product will be negative.

Step 5: Combine the Result

Combine the result from the multiplication with the negative sign obtained: \(-11.13\).

Key Concepts

absolute values

Absolute values represent the distance of a number from zero on the number line, regardless of its direction. For example, both \(-5.3\) and \(5.3\) have the same absolute value, \(5.3\). When multiplying signed numbers, it's essential to consider their absolute values first.

• The absolute value of a number is always positive.
• Denoted by vertical bars, e.g., \(|-7| = 7\).
• Simplifies multiplication by focusing only on the magnitude, not the sign.

In the exercise, we calculated \(5.3\) and \(2.1\) without their signs first.

negative numbers

Negative numbers are represented with a minus sign (-) and are located to the left of zero on the number line. They indicate a value less than zero. Understanding their behavior during multiplication is crucial:

• Multiplying two negative numbers yields a positive product. E.g., \((-3) \times (-2) = 6\).
• Multiplying a negative number by a positive number results in a negative product.
• Think of it as reversing direction: one negative reverses the direction once, another reversal makes it positive.

In our problem, \(-5.3 \times 2.1 = -11.13\) because a negative and a positive yield a negative result.

multiplication rules

The rules for multiplying signed numbers simplify the process and help avoid errors:

• \(+ \times + = +\).
• \(- \times - = +\).
• \(+ \times - = -\).
• \(- \times + = -\).

Following these rules ensures the correct sign of the product. Combining this with the multiplication of absolute values gives the final result accurately.
For example, in the given exercise, after calculating the absolute value product (11.13), we determined the sign based on the original numbers' signs, resulting in \(-11.13\).