Question

Students were given the task to fully simplify:
(2 to the power of 2 times 2 to the power of -6) to the power of -3 times (2) to the power of 0.
John believes the answer was 1 as any number to the power of 0 (except 0 to the power of 0) equals 1.
Carol believes that the answer is 4 to the power of 12 since 2 times 2 = 4 and for the exponents 2 + -6 = -4 and -4 times -3 = 12.
What is John's error?
What is Carol's error?
What is the correct answer? Show your work. (Do it correct. And please do not use brackets.)

Answer 1

To simplify the expression \((2^{2} \times 2^{-6})^{-3} \times 2^{0}\), we will break it down step by step.

First, let’s simplify the expression inside the parentheses:

1. **Combine the powers of \(2\)**:
\[
2^{2} \times 2^{-6} = 2^{2 + (-6)} = 2^{-4}
\]

2. **Now we raise this result to the power of \(-3\)**:
\[
(2^{-4})^{-3} = 2^{-4 \times -3} = 2^{12}
\]

3. **Next, we include \(2^{0}\)**:
\[
2^{12} \times 2^{0} = 2^{12 + 0} = 2^{12}
\]

Now let's summarize what we found:

The fully simplified expression is \(2^{12}\).

### Evaluation of John's and Carol's Errors:

- **John's Error**:
John mistakenly believes that the entire expression simplifies directly to 1 because he focuses only on \(2^{0}\) being 1. He ignores the contributions of the terms \(2^{2}\) and \(2^{-6}\) before applying exponent rules. Thus, he doesn’t correctly evaluate the full expression before concluding.

- **Carol's Error**:
Carol correctly combines \(2^{2} \times 2^{-6}\) to get \(2^{-4}\) but then makes an error by misapplying the exponent rules. Instead of treating \((2^{-4})^{-3}\) correctly to get \(2^{12}\), she computes \(-4 \times -3\) correctly but then incorrectly interprets the \(2^{12}\) as \(4^{12}\). The focus on \(4\) is a misuse of the base throughout the exponent computation.

### The Correct Answer:
The correct answer is:

\[
2^{12} = 4096
\]

Thus, the fully simplified form of the original expression is \(2^{12}\), which equals \(4096\).