Problem 28 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ (-7) 6 $$

Step-by-Step Solution

Answer

Answer: The value of the expression (-7)6 is -42.

Step 1: Identify the integers and their signs

The two integers are -7 and 6. The first integer -7 is negative and the second integer 6 is positive.

Step 2: Apply the multiplication rule for integers

According to the multiplication rule for integers, when a negative integer is multiplied by a positive integer, the result is a negative integer. Therefore, we will multiply the absolute values of the integers and then assign a negative sign to the result.

Step 3: Multiply the absolute values of the integers

The absolute value of -7 is 7, and the absolute value of 6 is 6. Multiply them together: $$ 7 \times 6 = 42 $$

Step 4: Assign the negative sign

Since the result should be negative as per the multiplication rule for integers, we assign a negative sign to the product obtained in Step 3: $$ (-7) \times 6 = -42 $$

Step 5: Write the final answer

The value of the expression (-7)6 is -42.

Key Concepts

Negative Numbers

Negative numbers are numbers less than zero. They are represented with a negative sign (-) in front of them. Understanding negative numbers is essential because:

• They indicate a decrease in value.
• They show opposite directions on a number line, such as below zero temperatures or debts.
• They are widely used in real-world applications, such as finance and science.

In challenges involving integer multiplication, identifying negative numbers allows us to apply specific rules easily. Since negative values behave differently from positives, recognizing their presence changes the operation outcomes.

Absolute Value

Absolute value refers to the distance a number is from zero on a number line, without regard to direction. Both negative and positive numbers can have absolute values.
Here’s what you need to know about absolute values:

• The absolute value of a number is always non-negative.
• For any number \( x \), the absolute value is denoted as \( |x| \).
• Example: \( |-7| = 7 \) and \( |6| = 6 \).

Knowing the absolute value lets us perform integer multiplication while temporarily ignoring the signs of the numbers, simplifying the calculation process.

Multiplication Rule

In mathematics, the multiplication rule for integers is crucial when dealing with both positive and negative numbers. When working with two integers, this rule helps us predict the sign of their product:

• If you multiply two positive numbers, the result is positive.
• If you multiply two negative numbers, the result is positive.
• If you multiply a negative number by a positive number, the result is negative.

This rule stems from the concept of direction on a number line. For example, when you perform \((-7) \times 6\), the rule recommends we multiply their absolute values and then assign the appropriate sign, leading to a correct solution.

Integer Operations

Integer operations include basic arithmetic functions such as addition, subtraction, multiplication, and division. These operations follow specific rules and guidelines:

• For addition and subtraction, combine like terms and carefully note the signs of the integers.
• For multiplication, apply the multiplication rules to establish the sign of the product.
• For division, consider the same rules as multiplication for determining the sign of the quotient.

Mastering integer operations is fundamental in mathematics, as they are often the building blocks of more complex mathematical problems and solutions.