Problem 36 - Fundamentals of Mathematics

Question

Use the order of operations to determine each value. \(\frac{\left(5^{2}-2^{3}\right)-2 \cdot 7}{2^{2}-1}+5 \cdot\left[\frac{3^{2}-3}{2}+1\right]\)

Step-by-Step Solution

Answer

21

Step 1: Evaluate Exponents

Evaluate the exponents first.\[5^2 = 25, \quad 2^3 = 8, \quad 2^2 = 4, \quad 3^2 = 9\]

Step 2: Perform Parenthesis Calculations

Calculate inside the parentheses and brackets.\[(5^2 - 2^3) = (25 - 8) = 17\]\[3^2 - 3 = 9 - 3 = 6\]

Step 3: Resolve Brackets and Division

Perform the division inside the brackets.\[\frac{3^2 - 3}{2} = \frac{6}{2} = 3\]Inside the brackets becomes:\[3 + 1 = 4\]

Step 4: Perform Multiplications

Perform the multiplications outside and within the operations.\[2 \cdot 7 = 14\]\[5 \cdot (3+1) = 5 \cdot 4 = 20\]

Step 5: Perform Subtractions and Final Operation

Complete the remaining operations in the original problem.The numerator:\[(17 - 14) = 3\]Now the denominator:\[(2^2 - 1) = 4 - 1 = 3\]Compute the overall expression:\[\frac{3}{3} + 20 = 1 + 20 = 21\]

Step 6: Final Result

The final value of the expression is 21.

Key Concepts

Exponents

Exponents are numbers raised to a power, indicating how many times to multiply the number by itself. In our problem, exponents appear frequently, and evaluating them correctly is crucial in the order of operations.
Let's break them down one by one:

• The expression \(5^2\) means 5 multiplied by itself, resulting in 25.
• Similarly, \(2^3\) means 2 multiplied three times: 2 x 2 x 2, which equals 8.
• \(2^2\) is 2 squared, resulting in 4.
• Finally, \(3^2\) is 3 squared, giving us 9.

Evaluating exponents early simplifies the expression and sets the stage for further calculations. Proper handling of exponents ensures accuracy throughout the problem.

Parentheses

Parentheses are used to group parts of an expression that need to be computed first. In our exercise, solving expressions within parentheses and brackets ensures that we respect their hierarchy within the order of operations.
After evaluating exponents, we perform operations inside parentheses:

• \((5^2 - 2^3)\) turns into \((25 - 8)\), resulting in 17.
• Inside the brackets, \(3^2 - 3\) simplifies to \(9 - 3\), giving 6.

Handling expressions inside parentheses or brackets first is essential. It simplifies the wider expression and helps avoid potential errors later in the problem.

Multiplication

Multiplication is distributing or scaling one quantity by another. In this problem, multiplication is an operation that needs to be handled after exponents and parentheses.
Let's identify and resolve multiplication:

• First, inside the main expression, we have \(2 \cdot 7\), which results in 14.
• In the bracketed section, we compute \(5 \cdot 4\), which yields 20.

Completing the multiplications allows us to narrow down the expression. This step is important because it typically transforms expressions into simpler forms, allowing for easier completion of subsequent operations.

Division

Division is distributing a quantity evenly and is often closely associated with multiplication in problems. Within our exercise, divisions occur after simplifying other components.
Here's how division is tackled in our expression:

• Within the brackets, we have \(\frac{6}{2}\), simplifying to 3. Coupled with 1, the content of the brackets simplifies to 4.
• Finally, the core expression's numerator is \((17 - 14)\), simplifying to 3, and the denominator is \((2^2 - 1)\), which results in 3.

This allows us to compute \(\frac{3}{3}\), turning into 1 after division. Understanding division's role in simplifying complex expressions is key to reaching accurate results.