Question

miguels steps in evaluating an expression are given below.

$32\div2(2^3 - 4)$

step 1: $= 32\div2(8 - 4)$
step 2: $= 32\div2(4)$
step 3: $= 32\div8$
step 4: $= 4$

what, if any, was miguels mistake?

\bigcirc miguel made no mistakes.
\bigcirc miguel incorrectly evaluated $2^3$ in step 1.
\bigcirc miguel incorrectly multiplied before dividing in step 3.
\bigcirc miguel should have used the distributive property in step 1.

Explanation:

Step1: Recall Order of Operations (PEMDAS/BODMAS)

The order of operations is Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). For the expression \(32\div2(2^{3}-4)\):

• First, handle the exponent inside the parentheses: \(2^{3}=8\) (Step 1 of Miguel does this correctly, so the option about Step 1 exponent is wrong).
• Then, simplify inside the parentheses: \(8 - 4=4\) (so Step 1 becomes \(32\div2(4)\), which is correct).

Step2: Analyze Multiplication/Division Order

Now, we have \(32\div2(4)\). Multiplication and division have the same precedence and are done from left to right. So we should do \(32\div2\) first, then multiply by 4? Wait, no—wait, the expression is \(32\div2\times4\) (since \(2(4)\) is \(2\times4\) when we consider order, but actually, the original expression is \(32\div2\times(2^{3}-4)\) simplified. Wait, no, let's re - express:

Wait, the expression is \(32\div2(2^{3}-4)\). After simplifying inside the parentheses, it's \(32\div2\times4\) (because \(2(4)\) is \(2\times4\), and division and multiplication are left - to - right). Wait, but Miguel in Step 3 does \(2\times4 = 8\) first, then \(32\div8\). But according to order of operations, we should do \(32\div2 = 16\) first, then \(16\times4=64\). Wait, no, wait—maybe I misread. Wait, the original expression: \(32\div2(2^{3}-4)\). Let's compute correctly:

1. Exponent: \(2^{3}=8\)
2. Parentheses: \(8 - 4 = 4\)
3. Now, the expression is \(32\div2\times4\) (because the \(2\) outside the parentheses is a multiplication, so it's \(32\div2\times4\)). Using left - to - right for multiplication/division: \(32\div2 = 16\), then \(16\times4 = 64\). But Miguel did \(2\times4 = 8\) first (multiplied before dividing), then \(32\div8 = 4\). So he incorrectly multiplied \(2\) and \(4\) before dividing \(32\) by \(2\). The distributive property is not required here (so the option about distributive property is wrong), and the exponent step was correct. So the mistake is in Step 3, where he multiplied before dividing (did \(2\times4\) before \(32\div2\)).

Answer:

Miguel incorrectly multiplied before dividing in step 3. (The option: Miguel incorrectly multiplied before dividing in step 3.)