Problem 88 - Intermediate Algebra

Question

Evaluate each expression. See Example \(9 .\) $$ \frac{2[-4-2(3-1)]}{3(3)(2)} $$

Step-by-Step Solution

Answer

The expression evaluates to 0.

Step 1: Simplify Inside the Parentheses

Begin by simplifying the expression inside the innermost parentheses: \[ 3 - 1 = 2 \]Now the expression looks like this:\[ \frac{2[-4-2 \times 2]}{3 \times 3 \times 2} \]

Step 2: Simplify Inside the Brackets

Continue simplifying the expression inside the brackets by performing the multiplication:\[ -2 \times 2 = -4 \]Substitute this value back into the expression:\[ \frac{2[-4 - (-4)]}{3 \times 3 \times 2} \]

Step 3: Solve the Expression Inside the Brackets

Next, resolve the expression inside the brackets:\[ -4 - (-4) = -4 + 4 = 0 \]Thus, the expression simplifies to:\[ \frac{2 \times 0}{3 \times 3 \times 2} \]

Step 4: Simplify the Multiplication and Division

Since the numerator is zero, the entire expression becomes zero, regardless of the denominator:\[ 0 = 0 \]

Step 5: Final Result

Hence, the evaluated result of the entire expression is:\[ 0 \]

Key Concepts

Order of Operations

When we are faced with a mathematical expression that includes multiple operations, it's crucial to perform these operations in the correct order. This established sequence of operations is commonly remembered by the acronym PEMDAS:

• Parentheses first
• Exponents (i.e., powers and roots, etc.)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

Consider PEMDAS as a set of rules that guide us to correctly simplify and solve expressions. Skipping or rearranging the order might lead to incorrect outcomes.
In the provided exercise, we first resolved operations inside parentheses, then multiplications, continuing in this logical sequence until we arrived at the final result.

Simplifying Expressions

Simplifying expressions is like cleaning up a messy desk—it makes the subsequent tasks much easier to handle. The goal here is to reduce complex expressions into a simpler, more manageable form by performing all possible operations.
In our example, we simplified step by step from the inside out, starting with operations inside the parentheses. By doing this, we gradually reduced the expression while maintaining its original value.
First, understand the power of simplifying: it allows for efficient calculations and minimizes errors in our computations. In the end, simplifying isn't just about getting to the answer more quickly—it's also about clarity and precision, making sure that each small part of the arithmetic aligns perfectly.

Parentheses in Mathematics

Parentheses are used to indicate which operations should be performed first in a mathematical expression. They function like a prioritized group of tasks.
In any equation, such as the one we solved here, handling what's inside parentheses first is critical. This ensures that interactions within the parentheses are completed before dealing with the rest of the expression, thereby adhering to the order of operations.
For example, in our problem, parentheses helped us recognize where to begin simplifying—inside \(3 - 1\). It's important to treat parentheses as a clear signpost that dictates an immediate action. Removing them too early or neglecting their priority can lead to errors, highlighting their essential role in mathematical clarity.