Problem 259 - Intermediate Algebra

Question

In the following exercises, multiply. $$ (-5.18)(-65.23) $$

Step-by-Step Solution

Answer

337.8914

Step 1: Understand the problem

The task is to multiply two negative numbers: \( (-5.18) \times (-65.23) \)

Step 2: Remember the rules

Multiplying two negative numbers results in a positive number.

Step 3: Perform the multiplication

Ignore the negative signs and multiply the absolute values: \( 5.18 \times 65.23 \).

Step 4: Calculate the product

Perform the multiplication: \( 5.18 \times 65.23 = 337.8914 \)

Step 5: Apply the sign

Since both original numbers were negative, their product is positive. Thus, \( (-5.18) \times (-65.23) = 337.8914 \)

Key Concepts

Absolute Values

Absolute values represent a number's distance from zero on the number line. It's always a positive number, regardless of the original number's sign. When dealing with absolute values, you remove any negative signs. For example, the absolute value of -5 is written as \(-5\) and equals 5.
In multiplication, using absolute values can simplify calculations. By focusing on the absolute values of the numbers involved, you can ignore their original signs until after the multiplication is performed.
When multiplying \(-5.18\) by \(-65.23\), the first step is to find their absolute values, which are 5.18 and 65.23 respectively. This simplification allows you to multiply these positive numbers directly, resulting in a more straightforward calculation.

Multiplication Rules

Multiplying numbers follows specific rules, particularly with positive and negative signs:

• Positive \( \times \) Positive = Positive
• Negative \( \times \) Positive = Negative
• Positive \( \times \) Negative = Negative
• Negative \( \times \) Negative = Positive

In this exercise, you have two negative numbers: \(-5.18\) and \(-65.23\). According to the multiplication rules, multiplying two negative numbers results in a positive number. This rule is essential because it ensures correct sign management in your calculations.
After computing the product of the absolute values, the final step is to apply the appropriate sign, determined by these rules. Thus, multiplying \(-5.18\) and \(-65.23\) gives a positive result.

Integer Operations

Understanding integer operations is key to solving various mathematical problems. Integers include positive numbers, negative numbers, and zero. When performing operations such as multiplication with integers, it's crucial to manage their signs and values appropriately.
Here's a quick guide to integer multiplication:

• Ignore the signs initially and multiply their absolute values.
• After obtaining the product, decide the final sign based on the multiplication rules.

The exercise example involves multiplying \(-5.18\) and \(-65.23\). First, ignore the negative signs and calculate the product of 5.18 and 65.23, which equals 337.8914. Then, apply the rule for two negative numbers, making the final result a positive 337.8914.
By understanding integer operations and their rules, you can easily tackle similar multiplication problems.