Question

Students were given the task to fully simplify: (2^2* 2^-6)^-3 * (2)^0










John believes the answer was 1 as any number to the zero power (except 0^0 ) equals 1.



Carol believes that the answer is 4^12 since 2*2=4 and for the exponents 2 + -6 = -4 and -4*-3=12




What is John's error?



What is Carol's error?



What is the correct answer. Show your work.

Answer 1

To simplify the expression \((2^2 \cdot 2^{-6})^{-3} \cdot (2)^0\), we'll go through the steps carefully.

### Step 1: Simplify the Inner Expression

First, we simplify the expression inside the parentheses:
\[
2^2 \cdot 2^{-6}
\]
Using the rule of exponents \(a^m \cdot a^n = a^{m+n}\), we can combine the exponents:
\[
2^{2 + (-6)} = 2^{-4}
\]

### Step 2: Apply the Outer Exponent

Next, we apply the exponent \(-3\) to the simplified expression:
\[
(2^{-4})^{-3}
\]
Using the power of a power rule, \( (a^m)^n = a^{m \cdot n} \):
\[
2^{-4 \cdot -3} = 2^{12}
\]

### Step 3: Include the \((2)^0\) term

Now, we consider the \((2)^0\) term:
\[
(2^{12}) \cdot (2^0)
\]
Using the same exponent rule \(a^m \cdot a^n = a^{m+n}\), we combine these:
\[
2^{12 + 0} = 2^{12}
\]

### Conclusion

So the entire expression simplifies to:
\[
2^{12}
\]

### Answer

The correct answer to the expression is:
\[
2^{12}
\]

### Evaluating Errors:
1. **John's Error**: John mistakenly thinks the entire expression evaluates to 1 based on \((2)^0\) being 1. However, he didn't account for the rest of the expression that doesn't evaluate to 0.

2. **Carol's Error**: Carol incorrectly interpreted the operations. While she correctly combined the exponents, she misapplied the rescaling of the result, suggesting \(4^{12}\) rather than recognizing the original base \(2\).

### Final Result
To present the final answer, if desired in numerical form:
\[
2^{12} = 4096
\]