Question

For Exercises \(79-86,\) explain the mistake in each problem and work it correctly. $$\begin{aligned} \frac{3^{2}-\left(25-4^{2}\right)}{2(-7)} & \\ =& \frac{3^{2}-(25-16)}{-14} \\ =& \frac{3^{2}-9}{-14} \\ =& \frac{9-9}{-14} \\ =& \frac{0}{-14}, \text { which is undefined } \end{aligned}$$

Short answer

Answer: 0

Step-by-step solution

Identify the Mistake

The mistake in this exercise is the assertion that 0 divided by -14 is undefined. In fact, zero divided by any non-zero number is always 0.

Correct the Mistake

The correct calculation for the expression \frac{0}{-14} should be 0. It should not result in 'undefined'.

Evaluate the entire expression correctly

Re-evaluate the given expression correctly from beginning to end as follows: First calculate the expression in the parenthesis (25 - 4^{2}), which leads to (25 - 16) = 9. Then subtract that result from 3^{2} resulting in 9 - 9 = 0. Lastly, divide that result by -14 to get 0. This leads to the following correct sequence of calculations: \begin{aligned} \frac{3^{2}-\left(25-4^{2}\right)}{2(-7)}\\ =& \frac{3^{2}-(25-16)}{-14} \\ =& \frac{9-9}{-14} \\ =& \frac{0}{-14} = 0 \end{aligned}

Key concepts

Order of Operations

In mathematics, the order of operations is a fundamental principle that dictates the sequence in which multiple operations should be performed to accurately evaluate an expression. It's known by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

When confronted with an expression like \(\frac{3^{2}-\left(25-4^{2}\right)}{2(-7)}\), it's important to follow these rules: begin by solving operations within parentheses (\(25-4^{2}\)), then exponents (\(3^2\) and \(4^2\)), followed by multiplication and division from left to right (in this case, the division by \(-14\)), and finally addition or subtraction if they were present.

Applying the correct order ensures that we arrive at the right result. In the exercise, the division comes last, after the subtraction has been completed within the parentheses and the exponents have been evaluated.

Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. These are the basic building blocks of all math problems and are often combined in various ways in expressions. In our example, we see a subtraction inside parentheses, an exponentiation, and a division.

To correctly work through these kinds of math problems, it is essential to perform each operation with care and precision. A mistake with any of these operations can lead to an incorrect result. It's particularly worth noting, as with the example division \(\frac{0}{-14}\), that division by zero is a common misconception. However, zero divided by any non-zero number results in zero, while division by zero itself is undefined.

Correcting Mistakes in Math

Mathematical proficiency isn't just about getting the right answer; it's also about identifying and correcting mistakes. In the exercise, the mistake was a misunderstanding of the division by zero rule. To correct mistakes in math, follow these steps:

• Review each step of the calculation carefully.
• Check the application of arithmetic operations and order of operations.
• Re-evaluate your computations to find where the error occurred.
• Correct the wrong operation as identified.
• Once corrected, go through the problem again to ensure accuracy.

Remember, identifying errors and understanding why they're wrong aids deeper learning and helps prevent similar mistakes in the future.

Evaluating Expressions

Evaluating expressions is the process of finding out what value an expression ultimately represents. This involves carrying out all operations in the expression following the order of operations.

In the exercise, evaluating the expression step-by-step with attention to detail and application of the order of operations led to correctly simplifying the given expression to zero. Precision in each step, from handling the exponents to properly carrying out the final division, is critical to evaluating expressions successfully.

Understanding that \(\frac{0}{-14} = 0\) is part of grasping how to evaluate divisions with zero. A zero in the numerator means the whole expression equals zero, assuming the denominator isn't also zero—which would make it undefined. Evaluating expressions correctly is a foundational skill in math that students need to develop for success in more complex math problems.