To solve the equation \(5x^{43} = 80\), we can first isolate \(x^{43}\):
\[ x^{43} = \frac{80}{5} \] \[ x^{43} = 16 \]
Next, we take the 43rd root of both sides to solve for \(x\). This can be written using a rational exponent:
\[ x = 16^{\frac{1}{43}} \]
Now, we need to evaluate \(16^{\frac{1}{43}}\). Since \(16\) is \(2^4\), we can rewrite it as:
\[ x = (2^4)^{\frac{1}{43}} = 2^{\frac{4}{43}} \]
This result does not correspond to any of the exact answers you provided, as none of the options (16, 40.32, 12, 8) are equal to \(2^{\frac{4}{43}}\). Therefore, it appears there might be an error in the provided choices or in my interpretation of your request.
If you intended to solve for \(x\) and select the closest rational number approximation, let’s calculate \(16^{\frac{1}{43}}\) using a calculator:
\[ 16^{\frac{1}{43}} \approx 1.436 \]
None of the original options (16, 40.32, 12, 8) are correct solutions. If the options are limited to integers, it's possible there has been a mistake in the formulation of the question or options provided.
If you'd like further clarification or assistance based on a different formulation, please let me know!
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