Problem 16 - College Algebra

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: Evaluate the exponent

You need to evaluate the expression inside the exponent first. Calculate \(3^2\): \[3^2 = 3 \times 3 = 9\]

Step 2: Perform the division

Next, perform the division operation in the expression. You have:\[18 \div 9 = 2\]

Step 3: Subtract the result from the previous step

Finally, subtract the result from Step 2 from 9:\[9 - 2 = 7\]Thus, the simplified expression is 7.

Key Concepts

Understanding Order of Operations

When solving algebraic expressions, it's essential to follow the order of operations. This ensures calculations are performed correctly. The order of operations is often remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In the expression you see, we first handle the exponent, then proceed with division, and lastly perform subtraction. The order is crucial as performing operations in the wrong order can lead to incorrect results. Each step builds on the previous result, guiding you to the right answer.

Exploring Exponents

Exponents are a way to express repeated multiplication compactly. When you see an exponent, it means you multiply the base by itself as many times as the value of the exponent. For example, in the expression \(3^2\),
the base is 3, and the exponent is 2, indicating that 3 should be multiplied by itself once, i.e., \(3 \times 3 = 9\).
This step is crucial in evaluating expressions because it lays the groundwork for other operations. Remember to always calculate the exponents before any multiplication, division, addition, or subtraction.

Decoding Division in Expressions

After handling exponents, the next operation in line is division. Division splits a number into equal parts. In the given expression, after evaluating the exponent, we perform the division operation found in the expression,
\(18 \div 9\). This means dividing 18 into 9 equal parts resulting in 2.
This step follows directly after evaluating the exponent because division has a higher precedence than subtraction. It helps simplify the expression further, allowing you to proceed to the final step.

Simplifying with Subtraction

Subtraction is the final operation in this sequence. It involves taking a number away from another. Once division has been completed, subtraction brings us to the final simplified form of the expression. In the exercise,
we see this as \(9 - 2\), where we subtract the result of the division from 9.
The difference, 7, is the simplified outcome of the original expression. Subtraction, as the final step, uses previous results and brings closure to simplifying the expression.