Problem 28 - Prealgebra

Question

Use any of the rules developed in this chapter and the rule for order of operations to simplify each of the following expressions as much as possible. [Examples 6–9] $$\frac{6(-3)}{-18}$$

Step-by-Step Solution

Answer

The expression simplifies to 1.

Step 1: Simplify the Numerator

In the expression \( \frac{6(-3)}{-18} \), we first simplify the numerator by multiplying the numbers 6 and \(-3\). This gives \(6 \times (-3) = -18\).

Step 2: Simplify the Fraction

Now the expression becomes \( \frac{-18}{-18} \). In this fraction, both the numerator and denominator are \(-18\).

Step 3: Apply the Division Rule

When dividing a number by itself (except zero), the result is 1. Since both the numerator and denominator are negative, a negative divided by a negative results in a positive, therefore, \( \frac{-18}{-18} = 1 \).

Key Concepts

Simplifying Expressions

Simplifying expressions involves breaking down complex mathematical statements into simpler forms, allowing for easier calculations and understanding. It often requires applying arithmetic operations, combining like terms, and following the order of operations (also known as PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
For the expression \( \frac{6(-3)}{-18} \), simplifying begins by addressing the operations in the numerator. This means multiplying 6 by \(-3\). Multiplying these gives \(-18\), which simplifies the expression to \( \frac{-18}{-18} \).
By simplifying the initial expression, we reduce its complexity, making it easier to apply further arithmetic rules like division.

Multiplication Rules

Multiplication is a key operation used in simplifying expressions. Understanding how to multiply signed numbers is crucial. Positive times positive or negative times negative equals a positive result, whereas positive times negative or negative times positive equals a negative result.

• Example: \(6 \times (-3) = -18\) indicates a positive multiplied by a negative, resulting in a negative value.
• Example: \((-3) \times (-3) = 9\) represents a negative multiplied by a negative, resulting in a positive value.

In our problem, the initial multiplication \(6 \times (-3)\) is straightforward. Recognizing these multiplication rules helps manage more complex mathematical expressions by establishing the sign of the product quickly.

Division Rules

Division rules are straightforward but essential, especially when dealing with signed numbers. The primary rule is that any number divided by itself equals 1, provided the number is not zero. Additionally, when dividing signed numbers, remember that a positive divided by a positive or a negative divided by a negative results in a positive quotient. Conversely, a positive divided by a negative or a negative divided by a positive results in a negative quotient.

• Example: \( \frac{9}{3} = 3\) and \( \frac{-9}{-3} = 3\).
• Example: \( \frac{9}{-3} = -3\) and \( \frac{-9}{3} = -3\).

In our simplified expression \( \frac{-18}{-18} \), since the numerator and denominator are equal and both negative, the division rule confirms that \( \frac{-18}{-18} = 1\). This demonstrates the principle that a negative divided by a negative becomes a positive, simplifying the whole expression to 1.