To solve the expression \( 75.03 - 3.4 \times (4.71 - 2.71)^4 \), we will follow the order of operations:
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Calculate the expression inside the parentheses: \[ 4.71 - 2.71 = 2.00 \]
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Raise the result to the power of 4: \[ (2.00)^4 = 2.00 \times 2.00 \times 2.00 \times 2.00 = 16.00 \]
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Multiply by 3.4: \[ 3.4 \times 16.00 = 54.4 \]
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Subtract from 75.03: \[ 75.03 - 54.4 = 20.63 \]
Thus, the final result is: \[ \boxed{20.63} \]