To simplify the expression \(\left(20a^{2}b^{-12}c^{-4}\right)\left(3a^{15}b^{12}c^{6}\right)\), we can start by multiplying the coefficients and then combine the like terms for the variables \(a\), \(b\), and \(c\).
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Multiply the coefficients: \[ 20 \times 3 = 60 \]
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Combine the \(a\) terms: \[ a^{2} \cdot a^{15} = a^{2+15} = a^{17} \]
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Combine the \(b\) terms: \[ b^{-12} \cdot b^{12} = b^{-12 + 12} = b^{0} = 1 \]
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Combine the \(c\) terms: \[ c^{-4} \cdot c^{6} = c^{-4 + 6} = c^{2} \]
Now, we combine all of these results together: \[ 60a^{17} \cdot 1 \cdot c^{2} = 60a^{17}c^{2} \]
Therefore, the simplified expression is: \[ \boxed{60a^{17}c^{2}} \]