Solve the following equation with a rational exponent: 2x32=16 .(1 point)

To solve the equation \(2x^{32} = 16\), we first rewrite 16 as a power of 2:

\[
16 = 2^4
\]

Now, we can rewrite the equation as:

\[
2x^{32} = 2^4
\]

Next, we can divide both sides by 2 (assuming \(2 \neq 0\)):

\[
x^{32} = 2^{4-1} = 2^3
\]

This simplifies to:

\[
x^{32} = 8
\]

Now we can express 8 as a power of 2:

\[
x^{32} = 2^3
\]

Since the bases are the same, we can equate the exponents:

\[
32 \log_2(x) = 3
\]

To solve for \(x\), we can express it in terms of a rational exponent:

\[
x = 2^{3/32}
\]

Calculating \(2^{3/32}\) can be approximated but doesn't directly match any of the four provided responses exactly. It's important to note that it does not resolve to any of the choices given (4, 8, no solution, 84). Therefore, the current responses do not match the solution derived from the equation.

Based on the choices presented and the solution we derived, the answer is **no solution**, as none of the other options are correct.