To solve the equation \( x^{23} = 64 \), we need to isolate \( x \). We can do this by applying both sides of the equation with the exponent \( \frac{1}{23} \).
Starting with the equation:
\[ x^{23} = 64 \]
We take the \( \frac{1}{23} \)th power of both sides:
\[ x = 64^{\frac{1}{23}} \]
Next, we express \( 64 \) in terms of a power of \( 2 \):
\[ 64 = 2^6 \]
Thus, substituting this into the equation gives:
\[ x = (2^6)^{\frac{1}{23}} = 2^{\frac{6}{23}} \]
We can convert this back to a numerical answer, but to find an approximate decimal value:
Calculating \( 2^{\frac{6}{23}} \) yields a number between \( 1 \) and \( 2 \).
To match your answer choices, we need to approximate:
- \( 8 = 2^3 \)
- \( 16 = 2^4 \)
- \( 32 = 2^5 \)
- \( 512 = 2^9 \)
Since \( \frac{6}{23} \) is less than \( 1 \), the value is closer to \( 2^0 = 1 \).
However, we cannot select one of the options accurately as no value matches directly; it appears none of the provided responses (512, 32, 16, 8) equals \( 2^{\frac{6}{23}} \).
Hence, since we are approximating a rational expression and the specific answer is not directly present in the options, if I had to choose one for a reasonable estimate, I would assess it but also on the basis that \( 2^{\frac{6}{23}} < 2 \).
If we must choose one from the original choices without matching exactly, we might opt for the response that represents numbers around \( 1 \) but choose not to ignore that the solution should rather lead us to look for \( 2^{\frac{6}{23}} \) being not equated fairly to any option therein.
In conclusion, the direct output of \( x \) from the equation is \( x = 2^{\frac{6}{23}} \), which does not match neatly with the selection given but leans towards less than \( 8 \).
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