Problem 65 - Elementary Algebra

Question

For exercises 1-80, evaluate. $$ \frac{12-(5-1)}{2^{2} \cdot 2} $$

Step-by-Step Solution

Answer

1

Step 1: Simplify Inside the Parentheses

Evaluate the expression inside the parentheses first: \(5 - 1 = 4\). This changes the expression to \( \frac{12-4}{2^{2} \, \cdot \, 2} \)

Step 2: Subtract Inside the Numerator

Perform the subtraction in the numerator: \(12 - 4 = 8\). This simplifies the expression to \( \frac{8}{2^{2} \, \cdot \, 2} \)

Step 3: Exponentiation in the Denominator

Evaluate the exponent in the denominator: \(2^{2} = 4\). This changes the expression to \( \frac{8}{4 \, \cdot \, 2} \)

Step 4: Multiplication in the Denominator

Multiply the numbers in the denominator: \(4 \, \cdot \, 2 = 8\). This now gives \( \frac{8}{8} \)

Step 5: Simplify the Fraction

Finally, simplify the fraction: \( \frac{8}{8} = 1 \)

Key Concepts

Parentheses

Handling parentheses is crucial in mathematics. It's the first step in the Order of Operations, often remembered by the acronym PEMDAS (Parentheses, Exponentiation, Multiplication, Division, Addition, Subtraction). Always start by solving expressions within parentheses. For instance, in the given problem, \( \frac{12 - (5 - 1)}{2^2 \cdot 2} \), we first calculate \(5 - 1 = 4\). This simplifies the expression to \( \frac{12 - 4}{2^2 \cdot 2} \). Remember, solving what's inside parentheses clears the path for the following operations.

Exponentiation

After parentheses, we move to exponentiation. Exponentiation means raising a number to a power. Here, we have \(2^2\), which means \(2 \cdot 2\). This results in \(4\). Therefore, our expression updates from \( \frac{8}{2^2 \cdot 2} \) to \( \frac{8}{4 \cdot 2} \). Understanding exponentiation helps you simplify complex expressions and is key to mastering many mathematical concepts.

Multiplication

Next in the Order of Operations is multiplication. Multiplying numbers is straightforward but can be tricky within complex expressions. In our expression, multiplication occurs as part of the denominator: \(4 \cdot 2\). Performing the multiplication gives us \(8\). So we simplify \( \frac{8}{4 \cdot 2} \) to \( \frac{8}{8} \). Accurate multiplication is essential for calculating the correct values in problems.

Subtraction

Subtraction is also an important step that occurs after parentheses and exponentiation. Here, it appears in the numerator: \(12 - 4\). Subtracting these numbers yields \(8\), reducing the expression to \( \frac{8}{4 \cdot 2} \). It's a simple process but crucial for achieving the correct final result. Subtraction can dramatically change the outcome if not performed accurately.

Fraction Simplification

The final concept is fraction simplification. Simplifying a fraction means reducing it to its simplest form. For \( \frac{8}{8} \), we simplify by noting that \(8 \div 8 = 1\). This gives us the final result: \(1\). Simplification helps in comparing, adding, or subtracting fractions and ensures clarity and simplicity in your answers. Always ensure your final fraction is in its simplest form.