To solve the equation \(2x^{32} = 16\), we can start by isolating \(x^{32}\).
1. Divide both sides by 2:
\[
x^{32} = \frac{16}{2}
\]
\[
x^{32} = 8
\]
2. Now we need to solve for \(x\). Take the 32nd root of both sides, which can be expressed using a rational exponent:
\[
x = 8^{\frac{1}{32}}
\]
3. We know that \(8\) can be expressed as \(2^3\):
\[
x = (2^3)^{\frac{1}{32}} = 2^{\frac{3}{32}}
\]
Now we will evaluate \(x\) to check if it matches any of the provided options:
- The options presented are: 8, 4, 84, and no solution.
Since \(2^{\frac{3}{32}}\) is a very small number (as \(3/32\) is less than 1), it will not match any of the whole numbers given in the options.
Thus, the correct choice from the options is:
**no solution**.