Problem 22 - College Algebra with Corequisite Support

Question

For the following exercises, simplify the given expression. \((12 \div 3 \times 3)^{2}\)

Step-by-Step Solution

Answer

The simplified expression is 144.

Step 1: Evaluate the Division

First, evaluate the expression inside the parentheses starting with division. Divide 12 by 3 to get 4. Now the expression becomes \((4 \times 3)^2\).

Step 2: Evaluate the Multiplication

Next, multiply the result from the previous step by 3. Multiply 4 by 3 to get 12. Now the expression becomes \(12^2\).

Step 3: Square the Result

Finally, square the result from the previous step. Compute the square of 12 to get \(12 \times 12 = 144\).

Key Concepts

Understanding the Order of Operations

When simplifying mathematical expressions, it's crucial to follow the order of operations. This helps ensure that everyone arrives at the same answer for the same problem. The standard rule for this is PEMDAS/BODMAS:

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

These steps dictate the sequence in which operations should be performed. In our exercise, since the division and multiplication are both inside the parentheses, you would handle these first, from left to right. Breaking the expression into clear steps prevents mistakes and clarifies your path to the solution.
It is often helpful to clearly note each consecutive operation performed according to these rules, which minimizes errors.

The Role of Exponents

Exponents represent how many times a number, called the base, is multiplied by itself. In an expression like \((12^2)\), the number 12 is squared, meaning you multiply 12 by itself once:

• 12 × 12

This results in 144. Understanding exponents is vital because they dramatically increase the value in an expression and are common in algebraic calculations.
In our exercise, the final step was squaring the result post multiplication, showcasing the power of exponents in changing expression outcomes significantly.
Mastery of exponents allows you to handle powers confidently and makes simplifying expressions more approachable.

Arithmetic Operations Simplified

Arithmetic operations form the backbone of simplifying expressions and involve the basic processes of addition, subtraction, multiplication, and division. Each operation has its own properties and affects numbers differently, making the orderly approach essential.

• Division is finding how many times a number is contained within another number.
• Multiplication involves combining equal groups, thus increasing a number’s value by repeated addition.

In our example, simplifying the expression starts with dividing 12 by 3, resulting in 4, then multiplying by 3 to get 12 again. Finally, you square the 12, which concludes with the value of 144.
This layered approach demonstrates how each operation builds on the previous result, highlighting the importance of accurate, step-by-step execution in arithmetic.