Question

select the correct answer.
a mistake was made in the steps shown to simplify the expression. which step includes the mistake?
\\(\frac{1 + 3^2}{5} + |-10| \div 2\\)
step 1: \\(= \frac{1 + 3^2}{5} + 10 \div 2\\)
step 2: \\(= \frac{1 + 9}{5} + 10 \div 2\\)
step 3: \\(= \frac{10}{5} + 10 \div 2\\)
step 4: \\(= 2 + 10 \div 2\\)
step 5: \\(= 12 \div 2\\)
step 6: \\(= 6\\)
\\(\bigcirc\\) a. step 4
\\(\bigcirc\\) b. step 3
\\(\bigcirc\\) c. step 1
\\(\bigcirc\\) d. step 5

Explanation:

Step1: Recall Order of Operations

The order of operations is PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) from left to right. We need to check each step against this.

Step2: Analyze Step 1

Simplify the absolute value: \(|-10| = 10\), so Step 1: \(\frac{1 + 3^3}{5}+10\div2\) is correct.

Step3: Analyze Step 2

Calculate the exponent: \(3^3 = 27\)? Wait, no, wait the original expression is \(3^3\)? Wait no, looking back, the original numerator is \(1 + 3^3\)? Wait no, the original expression is \(\frac{1 + 3^3}{5}\)? Wait no, in the problem, the first term is \(\frac{1 + 3^3}{5}\)? Wait no, looking at Step 2: \(\frac{1 + 9}{5}\). Wait, maybe it's \(3^2\) instead of \(3^3\)? Wait, the user's problem: let's recheck. The expression is \(\frac{1 + 3^3}{5}+|-10|\div2\)? Wait no, in Step 2, it's \(\frac{1 + 9}{5}\), so maybe it's \(3^2\) (since \(3^2 = 9\)). Let's assume that's a typo (maybe \(3^2\) instead of \(3^3\)) for the problem's sake. Then Step 2: \(1 + 9 = 10\)? Wait, \(1 + 9 = 10\), so \(\frac{10}{5}\) in Step 3. Then Step 4: \(\frac{10}{5}=2\), so Step 4: \(2 + 10\div2\) is correct. Then Step 5: The error is here. Because according to PEMDAS, we do division before addition. So \(10\div2 = 5\), then \(2 + 5 = 7\). But Step 5 does \(2 + 10 = 12\) then divides by 2, which is wrong. Wait, no, let's re-express each step:

Original expression: \(\frac{1 + 3^2}{5}+|-10|\div2\) (assuming \(3^2\) since Step 2 has 9).

Step 1: \(\frac{1 + 3^2}{5}+10\div2\) (correct, absolute value)

Step 2: \(\frac{1 + 9}{5}+10\div2\) (correct, \(3^2 = 9\))

Step 3: \(\frac{10}{5}+10\div2\) (correct, \(1 + 9 = 10\))

Step 4: \(2 + 10\div2\) (correct, \(\frac{10}{5}=2\))

Step 5: Here, the mistake is. Because we must do division before addition. So \(10\div2 = 5\), so we should do \(2 + 5\), not \(2 + 10\) then divide by 2. So Step 5: \(12\div2\) is wrong. Wait, but let's check the options. Option D is Step 5. Wait, but let's confirm:

Wait, the operations: after Step 4: \(2 + 10\div2\). According to PEMDAS, division comes before addition. So \(10\div2 = 5\), then \(2 + 5 = 7\). But Step 5 does \(2 + 10 = 12\) (adding first) then \(12\div2 = 6\), which violates PEMDAS. So Step 5 is incorrect.

Wait, but let's check each option:

A. Step 4: \(2 + 10\div2\) is correct (since \(\frac{10}{5}=2\))

B. Step 3: \(\frac{10}{5}\) is correct (1 + 9 = 10)

C. Step 1: Correct (absolute value)

D. Step 5: Incorrect, because we should do division before addition. So \(10\div2 = 5\), then \(2 + 5 = 7\), not \(12\div2\). So Step 5 is wrong.

Answer:

D. Step 5