To solve the equation \( x^{23} = 64 \), we first need to express 64 as a power of a number. We know that:
\[ 64 = 2^6 \]
Now, we can rewrite the equation as:
\[ x^{23} = 2^6 \]
To solve for \( x \), we will take the \( 23^{rd} \) root of both sides:
\[ x = (2^6)^{\frac{1}{23}} = 2^{\frac{6}{23}} \]
This is the exact value for \( x \). We can also express this in decimal or approximate forms if needed, but if you are looking for rational numbers, \( 2^{\frac{6}{23}} \) is a valid answer.
Since the responses provided are integer solutions, we should evaluate \( 2^{\frac{6}{23}} \). However, this value may not correspond with any of the integer options you provided (8, 32, 16, 512).
Now let's explore a bit more: If you were looking for possible integer solutions where \( x \) might be interpreted differently or if the equation might imply estimates, Satisfying integer options under the constraints of rational exponent should be checked carefully.
- 8: \( 8^{23} = 2^{3 \cdot 23} = 2^{69}\) (not 64)
- 32: \( 32^{23} = 2^{5 \cdot 23} = 2^{115}\) (not 64)
- 16: \( 16^{23} = 2^{4 \cdot 23} = 2^{92}\) (not 64)
- 512: \( 512^{23} = 2^{9 \cdot 23} = 2^{207}\) (not 64)
None of the provided options satisfy the original equation correctly, hence the answer \( x = 2^{\frac{6}{23}} \) remains the most logical resolution of the equation. If we are supplied with additional context, please revise your query with more directives, or the frameworks under which the question is posed.
Given the integer constraints outlined in the responses, none align accurately with the derived solution from the rational exponent equation. Thus, no precise integer solution from the options reflects the original problem's request accurately.
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