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

Question

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: \[ 2^{12} = 4096 \]