Problem 22 - College Algebra

Question

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

Step-by-Step Solution

Answer

The simplified expression is 144.

Step 1: Simplify Inside the Parentheses

First, solve the expression inside the parentheses. The expression is \(12 \div 3 \cdot 3\), which follows the order of operations (PEMDAS/BODMAS). Divide first: \(12 \div 3 = 4\). Then multiply: \(4 \cdot 3 = 12\). Now the expression inside the parentheses is simplified to 12.

Step 2: Apply the Exponent

Once the expression inside the parentheses is simplified to 12, apply the exponent. Raise 12 to the power of 2: \(12^2 = 144\).

Key Concepts

Simplifying Expressions

Simplifying expressions is like tidying up a messy room. We want to take a complex expression and turn it into its simplest form. This is achieved by following specific rules, including the order of operations. When simplifying, you shouldstart with the operations inside parentheses and then proceed to exponents, multiplication, division, addition, and subtraction. In our example, 1. The expression inside the parentheses, \(12 \div 3 \cdot 3\), is first simplified by performing division and multiplication.2. This follows the PE(MD)AS rule, where after parentheses, multiplication and division are handled from left to right.3. Simplifying inside the parentheses gives us 12, which is then subject to the exponent operation.

• Breaking down complex expressions helps in solving them easily.
• Always follow the order of operations to avoid mistakes.

These steps ensure that anyone can confidently simplify mathematical expressions, step by step.

Exponents

Exponents represent the number of times a number, known as the base, is multiplied by itself. In the expression 1. \(12^2\), 12 is the base and 2 is the exponent.2. When we say \(12^2\), it means 12 is multiplied by itself: \(12 \times 12\).Using exponents, we can write repeated multiplication in a compact way. For instance, rather than writing \(12 \times 12\), we simply write \(12^2\).

• Exponents make calculations surprisingly simple and neat.
• It's always important to handle exponents carefully after simplifying inside any parentheses.

This makes expressions easier to understand and work with.

Parentheses in Math

Parentheses in math are like brackets in a written sentence. They show you which part of the equation to solve first. The order of operations dictates that you always resolve expressions inside parentheses before anything else. In our example,1. \((12 \div 3 \cdot 3)\) highlights the part of the expression that should be simplified right away.2. Notice that the parentheses ensure the division and multiplication inside take precedence over the exponent.By focusing on the operations inside the parentheses first, we prevent order of operation mistakes that could otherwise lead to incorrect solutions. That is one reason why understanding the role and placement of parentheses is crucial in mathematics.

• They dictate the sequence of calculations within an expression.
• Misplacing them can change the outcome entirely, so use them wisely while setting up your expression.

Starting with parentheses sets the stage for accurate problem-solving.