Problem 16 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the given expression. \(9-18 \div 3^{2}\)

Step-by-Step Solution

Answer

The simplified expression is 7.

Step 1: Identify the Order of Operations

In mathematics, the order of operations must be followed to correctly simplify expressions. This order can be remembered using the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In this expression, there are exponents and division that need to be handled first before subtraction.

Step 2: Simplify the Exponent

According to PEMDAS, evaluate the exponent first. In the expression, the exponent is \(3^2\), so calculate \(3^2 = 9\).

Step 3: Divide

Next, perform the division. Substitute \(3^2\) with 9, and divide: \(18 \div 9 = 2\).

Step 4: Subtract

Finally, subtract the result from the division (which is 2) from 9 as follows: \(9 - 2 = 7\).

Key Concepts

PEMDAS

In mathematics, following the order of operations is crucial to simplify expressions accurately. This order is captured by the acronym **PEMDAS**:

• Parentheses: Solve expressions inside parentheses first.
• Exponents: Next, deal with exponents, or powers.
• Multiplication and Division: Perform these operations from left to right as they appear in the equation.
• Addition and Subtraction: Lastly, tackle these from left to right.

For our expression, **9** - **18 ÷ 3²**, we need to pay attention to exponents and division first, according to PEMDAS, before dealing with any subtraction. Remembering this order ensures we don't make mistakes in simplifying expressions.

Exponents

In the acronym PEMDAS, the **E** stands for "Exponents." An exponent refers to the number of times a number is multiplied by itself. For example, in our expression, we have the term **3²**. This notation means 3 multiplied by itself, which is calculated as:\[ 3^2 = 3 \times 3 = 9 \]Dealing with exponents early prevents errors later in simplifying the expression. Exponents are a powerful part of algebra and are critical in equations, so mastering them will help you in many areas of math and science. Once the exponent has been simplified to **9**, we can proceed to the next step: division.

Division

Next in the order of operations comes division, represented by **D** in PEMDAS. When you see division in an expression, you perform it after calculating any exponents, but before handling addition or subtraction. In the expression **9 - 18 ÷ 3²**, once we simplified **3²** to 9, our next step is division:\[ 18 \div 9 = 2 \]Here, we divide 18 by 9 to get 2. Division simplifies expressions into smaller parts, making them easier to handle. It’s a fundamental operation you’ll use in many other areas of math, so ensure you're comfortable with it. Once division is done, all that remains is the straightforward subtraction.

Subtraction

Finally, we arrive at subtraction. It’s the **S** in PEMDAS and is typically handled last after all parentheses, exponents, multiplication, and division have been completed. In our expression, **9 - 18 ÷ 3²**, we’ve already simplified division to get 2. This leaves us with:\[ 9 - 2 = 7 \]Subtraction reduces a number by another, and it’s one of the basic tools of arithmetic you use daily. Practice helps you calculate faster and catch errors early. With subtraction complete, the final simplified result of our original expression is 7, demonstrating how each step in PEMDAS effectively organizes and solves the problem.