Question

Students were given the task to fully simplify
(2^2•2^-6)^-3•(2)^0
John believes the answer was one as any number to the zero power except zero to the power of zero equals one. Carol believes the answer is four to the power of $.12 2×2 equals equals four and for the exponents 2+ -6 equals -4 and -4 times -3 equals 12
What is John’s error?
What is Carol’s error?
What is the correct answer?
Show your work

Answer 1

Let's break down the expression \((2^2 \cdot 2^{-6})^{-3} \cdot (2)^0\) step by step and identify the errors made by John and Carol.

First, let's simplify the expression:

### Step 1: Simplify inside the parentheses
\[
2^2 \cdot 2^{-6} = 2^{2 + (-6)} = 2^{-4}
\]

### Step 2: Raise to the power of -3
Now we need to simplify \((2^{-4})^{-3}\):
\[
(2^{-4})^{-3} = 2^{-4 \cdot -3} = 2^{12}
\]

### Step 3: Simplify \(2^0\)
Now consider the other part of the expression, which is \( (2)^0 \):
\[
2^0 = 1
\]

### Step 4: Multiply the results together
Now combine the two parts:
\[
2^{12} \cdot 1 = 2^{12}
\]

### Step 5: Final answer
Thus, the correct answer is:
\[
2^{12} = 4096
\]

### Identifying Errors

**John's Error:**
John incorrectly believed that the entire expression simplifies to 1 simply because of the \( (2)^0 \) term. However, he failed to recognize that the other part of the expression, \( (2^2 \cdot 2^{-6})^{-3} \), is not zero and indeed contributes significantly to the final result. The zero exponent only applies to \( 2^0 = 1 \).

**Carol's Error:**
Carol made a mistake in simplification. While she correctly recognized that \(2^2 \cdot 2^{-6} = 2^{-4}\), she miscalculated the powers during further steps. Specifically, she misinterpreted the calculations of powers:
- She simplified the expression as if \( -4 \) was added directly to \( -3 \) rather than multiplying. The correct multiplication should yield \(2^{12}\) rather than \(2^0\) after the negative exponent simplification.

### Conclusion
The correct answer is:
\[
2^{12} = 4096
\]