To simplify the expression \(-25x^8 \cdot 5x\), we can apply the properties of exponents and the associative property of multiplication.
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Multiply the coefficients: Multiply the numerical coefficients \(-25\) and \(5\): \[ -25 \cdot 5 = -125 \]
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Apply the properties of exponents: When multiplying base \(x\) with the same base, we add the exponents. The exponents here are \(8\) for \(x^8\) and \(1\) for \(x\) (since \(x\) can be written as \(x^1\)): \[ x^8 \cdot x^1 = x^{8+1} = x^9 \]
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Combine the results: Now, we can combine the results from the two steps above: \[ -25x^8 \cdot 5x = -125x^9 \]
Thus, the simplified expression is: \[ \boxed{-125x^9} \]