Problem 8 - Algebra A Combined Function

Question

Multiply. \(-2 \cdot 37\)

Step-by-Step Solution

Answer

The product of (-2) imes 37 = -74.

Step 1: Identify the numbers and signs

We have the numbers (-2) and 37. We will need to multiply these two numbers together. Notice that (-2) is negative and 37 is positive.

Step 2: Multiply the absolute values

First, take the absolute values of the numbers, which are 2 and 37. Now multiply them: 2 imes 37 = 74.

Step 3: Apply the sign rule for multiplication

The rule for multiplication with negative numbers states that a negative number multiplied by a positive number results in a negative product. Therefore, the product will be -74.

Step 4: Write the final result

Considering the sign rule and the multiplication result, (-2) imes 37 = -74.

Key Concepts

Understanding Multiplication Rules

Multiplying integers involves a few rules that are easy to remember and apply. Here's a straightforward way to understand them:

• If you multiply two positive numbers, the result is positive. For example, \( 3 \times 4 = 12 \).
• Similarly, multiplying two negative numbers results in a positive product, such as \((-5) \times (-6) = 30 \). We multiply their absolute values and because two negatives make a positive, the product is positive.
• However, when multiplying a positive number by a negative number, or vice versa, the product is negative.For instance, the example given \((-2) \times 37 = -74 \) utilizes this rule. The reason is simple: a negative times a positive gives a negative.

These rules are the foundation for integer multiplication and help simplify problems.

Working with Negative Numbers

Negative numbers are essential in solving many real-world problems, and understanding how to work with them is crucial in mathematics. A negative number is any number that is less than zero. These numbers are often found in temperature (below zero degrees), finances (debt), and altitude (below sea level).When multiplying with negative numbers, the sign of the product depends on the signs of the involved numbers:

• Negative times positive is negative. This is seen in \((-2) \times 37 \), resulting in \(-74 \).
• Negative times negative equals positive, as two negatives cancel out each other's effects on the sign.

The negative symbol merely indicates direction or position in mathematics, allowing numbers to be more versatile by expanding their application beyond simply counting.

Exploring Absolute Values

Absolute values are a key concept that helps simplify problems involving negative numbers. The absolute value of a number is its distance from zero on a number line, without considering direction. Thus, both \( +3 \) and \(-3 \) have the same absolute value, which is 3.When performing operations like multiplication:

• First find the absolute values of the numbers involved. For example, in \((-2) \times 37\), the absolute values are \(2\) and \(37\).
• Multiply these absolute values first, then apply the appropriate sign based on the multiplication rules for positive and negative numbers.

This approach ensures the calculations are uncomplicated and helps keep track of the signs, which can alter the result significantly.