To solve the equation \(5x^{43} = 80\), we will isolate \(x^{43}\) first.
- Divide both sides by 5:
\[ x^{43} = \frac{80}{5} \] \[ x^{43} = 16 \]
- Now, to solve for \(x\), we can take the \(43\)-rd root of both sides, which can also be expressed using a rational exponent:
\[ x = 16^{\frac{1}{43}} \]
Next, we need to calculate \(16^{\frac{1}{43}}\). Since \(16\) is \(2^4\):
\[ 16^{\frac{1}{43}} = (2^4)^{\frac{1}{43}} = 2^{\frac{4}{43}} \]
However, to find a numerical approximation or recognize if one of the provided choices is \(2^{\frac{4}{43}}\), we can check if it fits.
Evaluating \(2^{\frac{4}{43}}\):
This is a small number since the exponent \(\frac{4}{43}\) is less than 1. In context to your options—40.32, 16, 12, and 8—none seem to relate directly to \(2^{\frac{4}{43}}\).
To sum up, the solution \(x = 16^{\frac{1}{43}}\) doesn't match the provided responses (likely indicating a misunderstanding in the interpretation of solutions). If we were to take into account the potential answer options, the only mathematical solution should be \(x \approx 2^{\frac{4}{43}}\), which is a very small value and doesn't directly yield any of the integer options provided.
Thus, based on the solution method, none of the presented responses are numerically valid as they do not correspond to \(x\). For the proper rounded figure or understanding within the parameters of common numerical values, please ensure the original equation and options are correlated and correctly transcribed based on the problem's requirements.
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