Solve the following equation with a rational exponent: x23=64 .(1 point)

To solve the equation \( x^{23} = 64 \), we will first express 64 as a power of 2:

\[
64 = 2^6
\]

Now we can rewrite the equation:

\[
x^{23} = 2^6
\]

To solve for \( x \), we take the 23rd root of both sides:

\[
x = (2^6)^{\frac{1}{23}} = 2^{\frac{6}{23}}
\]

Now, we can determine the numerical value. However, since none of the provided responses (512, 32, 16, 8) are in the form \( 2^{\frac{6}{23}} \), let's evaluate \( x \) to find an approximate numerical value of \( x \).

Calculating \( 2^{\frac{6}{23}} \) using a calculator gives a number around 1.440. Therefore, the closest whole numbers from the responses provided appear more likely to be estimations or require different contexts for rounding.

Given the options, note that \( x = 2^{6/23} \) is less than 2 but more than 1. Based on the provided options:
- 512 \( (2^9) \)
- 32 \( (2^5) \)
- 16 \( (2^4) \)
- 8 \( (2^3) \)

All those values are much larger than \( 2^{6/23} \).

If this problem sought just rational approximations or rounding:
- We would still identify that the solution neither fits exactly into one of those provided output options.

Thus, it seems there is some misunderstanding with the options reflecting the solution or evaluation necessary. The answer \( x = 2^{\frac{6}{23}} \) is the formal resolution to the core equation \( x^{23} = 64 \). If you'd like clarification on approximate answers among broader interpretations, please clarify what interpretations might fit from the multiple choice.