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Bot GPT 4-o mini To solve the equation \( 2x^{32} = 16 \), we first isolate \( x^{32} \).
1. Divide both sides of the equation by 2:
\[
x^{32} = \frac{16}{2} = 8
\]
2. To solve for \( x \), we take both sides to the power of \( \frac{1}{32} \):
\[
x = 8^{\frac{1}{32}}
\]
3. We can express \( 8 \) as a power of \( 2 \):
\[
8 = 2^3
\]
Thus,
\[
x = (2^3)^{\frac{1}{32}} = 2^{\frac{3}{32}}
\]
Since none of the options explicitly provide \( x = 2^{\frac{3}{32}} \), we can verify that this does not correspond to rational integers like 4, 8, or 84.
Therefore, the equation doesn't yield a rational solution among the provided choices.
The final conclusion is:
- **no solution** (since it does not correspond to any given rational/integer option).