Problem 22 - College Algebra

Question

Simplify the given expression. $$ (12 \div 3 \cdot 3)^{2} $$

Step-by-Step Solution

Answer

144

Step 1: Perform Division

First, simplify the expression inside the parentheses by performing the division operation. This involves dividing 12 by 3: \[12 \div 3 = 4\] The expression is now: \[(4 \cdot 3)^{2}\]

Step 2: Perform Multiplication

Next, multiply the result from the previous step by 3, as indicated within the parentheses: \[4 \cdot 3 = 12\] The expression now simplifies to: \[12^{2}\]

Step 3: Apply Exponentiation

Finally, apply the squared exponent by multiplying 12 by itself: \[12^{2} = 144\] Therefore, the expression simplified to its final value: 144.

Key Concepts

Order of Operations

The order of operations is a key concept in algebra that ensures consistency when solving mathematical expressions. When faced with complex equations, following the order of operations ensures you arrive at the correct solution.
The correct order of operations can be remembered by the acronym PEMDAS, which stands for:

• P - Parentheses
• E - Exponents (or powers)
• MD - Multiplication and Division (from left to right)
• AS - Addition and Subtraction (from left to right)

For the expression given above, we start inside the parentheses. Performing division first (12 divided by 3) and then multiplication helps maintain order within the expression. By always following PEMDAS, you can tackle algebraic expressions systematically and avoid mistakes.

Exponentiation

Exponentiation, in simple terms, means raising a number to the power of another, which involves multiplying the number by itself a certain number of times. For example, in the expression \(12^2\), the exponent is 2, indicating that 12 should be multiplied by itself once, resulting in 144.
Let's go over a few key points about exponents:

• Exponents are written as superscripts, a small number to the upper right of a base number.
• The base is the larger number, and it is the number that gets multiplied.
• Exponentiation is distinct from multiplication as it involves repeated multiplication.

Understanding exponentiation is crucial in algebra as it appears frequently in equations, allowing you to simplify expressions efficiently and effectively.

Simplification

Simplification in algebra is about reducing complex expressions to their simplest form while maintaining the same value. This process often involves using operations such as addition, subtraction, multiplication, division, and exponentiation.
Why is simplification important?

• It makes complex algebraic expressions easier to understand and work with.
• Simplified expressions can reveal underlying patterns or properties that might not be obvious in their original form.
• Simplification is crucial for solving algebraic equations, allowing the variable's value to be found more easily.

During simplification, such as in the given exercise, we systematically apply the order of operations. We first simplify the operations within parentheses, then handle any multiplication or division, and finally apply exponentiation. This step-by-step approach helps us break down the expression effectively, yielding a final, simplified answer.