Problem 68 - Intermediate Algebra

Question

Evaluate each expression. See Example \(8 .\) $$ -18 \div 6 \cdot 3 $$

Step-by-Step Solution

Answer

The value of the expression is -9.

Step 1: Understand the Order of Operations

The order of operations dictates that you perform division and multiplication from left to right. For the expression \(-18 \div 6 \cdot 3\), we identify the operation to perform first, which is the division.

Step 2: Perform the Division

Calculate \(-18 \div 6\). Dividing \(-18\) by \(6\) gives \(-3\). So, the expression simplifies to \(-3 \cdot 3\).

Step 3: Perform the Multiplication

Multiply \(-3\) by \(3\). This results in \(-9\), as multiplying a negative number by a positive number yields a negative product.

Key Concepts

Division and Multiplication

When approaching expressions that involve both division and multiplication, it's crucial to follow the order of operations. This means you tackle these operations as they appear from left to right.

Let’s take the example from our problem:

• First, we encountered the division: \(-18 \div 6\).
• This simplifies to \(-3\), as dividing a negative number by a positive number results in a negative number.
• Next, you perform the multiplication step: \(-3 \cdot 3\).
• This product is \(-9\).

Multiplication and division are performed with equal priority, and you process them as they appear, moving left to right across the expression. Following this method helps ensure that you arrive at the correct final answer. Remember: left-to-right priority in the absence of parentheses or other grouping symbols.

Negative Numbers

Negative numbers can make calculations feel a bit trickier, but understanding how they behave in operations is essential. When you divide or multiply negative numbers, applying the correct rules is key to avoiding mistakes.

Here are some points to remember about negative numbers:

• Division: When dividing a negative by a positive, like \(-18 \div 6\), the quotient is negative, giving us \(-3\). In general, dividing numbers with differing signs yields a negative result.
• Multiplication: If you multiply a negative number by a positive number, such as in \(-3 \cdot 3\), the result is also negative, resulting in \(-9\). However, if both numbers were negative, the product would be positive.

These basic rules help guide you through expressions involving negative numbers without getting tripped up by the signs.

Simplifying Expressions

Simplifying expressions is the art of reducing them to their simplest form. For the task at hand, it means performing all the operations methodically until you can't simplify anymore.

In our example:

• We began with \(-18 \div 6 \cdot 3\).
• We first simplified by dividing, giving \(-3\).
• Then we moved on to multiply, reaching the simplified expression of \(-9\).

Each step moves you closer to a single, concise answer. Simplifying is about breaking down complex operations into simpler parts, and using the order of operations to guide you ensures clarity and accuracy. In every simplification process, patience and attention to each step are key.