To solve the equation \( 2x^{32} = 16 \), we can start by rewriting 16 as a power of 2.
Since \( 16 = 2^4 \), we can rewrite the equation as:
\[
2x^{32} = 2^4
\]
Next, we can divide both sides of the equation by 2:
\[
x^{32} = 2^{4-1} = 2^3
\]
Now we have:
\[
x^{32} = 8
\]
To solve for \( x \), we raise both sides to the power of \( \frac{1}{32} \):
\[
x = 8^{\frac{1}{32}}
\]
Since \( 8 = 2^3 \), we can rewrite this as:
\[
x = (2^3)^{\frac{1}{32}} = 2^{\frac{3}{32}}
\]
This gives us a real number value for \( x \) instead of one of the provided integer choices. However, let's approximate it.
Since \( 2^{\frac{3}{32}} \) is between 1 and 2, we can conclude that \( x \) does not match any of the integer options given (4, 8, 84, or no solution) precisely.
Thus, the answer is **no solution** within the context of the integer responses provided.