Problem 124 - Elementary Algebra

Question

Problem: Evaluate: \(4+(7-1)^{2}\) Incorrect Answer: \(4+(7-1)^{2}\) $$ \begin{aligned} &=4+6^{2} \\ &=4+12 \\ &=16 \end{aligned} $$

Step-by-Step Solution

Answer

The correct answer is 40.

Step 1: Simplify Inside the Parentheses

First, evaluate the expression inside the parentheses: (7 - 1) = 6.

Step 2: Apply the Exponent

Next, apply the exponent to the simplified result inside the parentheses: 6^{2} = 36.

Step 3: Add the Remaining Number

Finally, add the result from the exponentiation to the number outside the parentheses: 4 + 36 = 40.

Key Concepts

Parentheses

Parentheses are used in mathematical expressions to group parts of the equation that should be evaluated first according to the order of operations. It's like giving priority to certain calculations so they are done before the others.

Here's how you handle parentheses:

• Look inside the parentheses and perform all operations there first.
• Always resolve expressions inside nested parentheses (parentheses within parentheses) starting from the innermost ones.

For example, in the given exercise, the expression inside the parentheses is \(7 - 1\). By performing the subtraction first, we get the result of 6. Only then we move on to other operations outside the parentheses.

Exponents

Exponents tell us how many times to multiply a number by itself. They are a critical part of the order of operations and come after you have simplified the parentheses.

To apply exponents correctly:

• First, calculate any operations inside parentheses.
• Then, apply the exponent to the simplified number or expression.

Using our example, once we found that \(7 - 1 = 6\), we applied the exponent. So, \(6^2 = 6 \times 6 = 36\). This step is completed before you proceed to addition or any other operations in the expression.

Addition

Addition is one of the simplest operations but it must always follow the correct order of operations. That means once you've dealt with parentheses and exponents, you can finally move on to adding numbers.

To properly handle addition:

• Make sure all operations inside parentheses and exponents are resolved first.
• Add the results from these operations to any other numbers left in the expression.

In our example, after simplifying the expression inside the parentheses to get 6 and then squaring it to get 36, we finally add the 4: \(4 + 36 = 40\). This gives us the correct answer to the problem.