Problem 34 - Beginning and Intermediate Algebra

Question

The value of an expression was found incorrectly as follows. $$\begin{aligned}16 &-2^{3}+5 \\\&=16-6+5 \\\&=10+5 \\\&=15\end{aligned}$$ Find the correct value of the expression.

Step-by-Step Solution

Answer

The correct value of the expression is 13.

Step 1: Identify the given problem

The expression given is \[16 - 2^{3} + 5\] and it was incorrectly evaluated to 15.

Step 2: Understand the order of operations

Recall the order of operations, PEMDAS/BODMAS, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

Step 3: Calculate the exponent first

Evaluate the exponent first. Calculate \(2^{3}\):\[2^{3} = 8\]

Step 4: Substitute the exponent in the expression

Replace the exponent term in the original expression with its calculated value:\[16 - 8 + 5\]

Step 5: Perform the subtraction

Next, perform the subtraction operation:\[16 - 8 = 8\]

Step 6: Perform the addition

Lastly, perform the addition operation:\[8 + 5 = 13\]

Step 7: Verify the final answer

The final result after performing all operations correctly is 13.

Key Concepts

Understanding PEMDAS

When we solve mathematical expressions, it's important to follow the right order. This order is called PEMDAS. PEMDAS stands for:

• P: Parentheses
• E: Exponents
• M: Multiplication
• D: Division
• A: Addition
• S: Subtraction

These rules help us determine which operations to perform first.
For example, in the expression \(16 - 2^3 + 5\), we first handle any exponents before we perform addition or subtraction. Ignoring the order can result in incorrect answers, like the 15 in the original problem.

Working with Exponents

Exponents indicate that a number should be multiplied by itself a certain number of times. For example, \(2^3\) means multiplying 2 by itself three times: \(2 * 2 * 2 = 8\).
Calculating exponents first is crucial. In our example expression \(16 - 2^3 + 5\), we found that \(2^3 = 8\). Only after this calculation, we replace \(2^3\) with 8 in the expression. This simplifies our expression to \(16 - 8 + 5\).

Getting Subtraction Right

Subtraction is the process of taking one number away from another. It’s important to perform subtraction in the correct sequence according to PEMDAS.
In the expression \(16 - 8 + 5\), after calculating the exponent, we next perform the subtraction \(16 - 8\). This results in 8, giving us a new expression: 8 + 5. Incorrectly skipping steps can lead to the wrong answer.

Simple Addition Explained

Addition sums numbers together. Using PEMDAS, we often perform addition last unless parentheses or exponents change the order.
In our expression, we finally add the remaining terms: \(8 + 5\). Calculating this, we find \(8 + 5 = 13\). Addition is straightforward, but it's critical to follow the sequence steps correctly to arrive at this final step.