Solve the exponential equation without using logarithms. \[ 4^{-x}=16^{x+12} \] \[ x=\square \text { (Simplify your answer.) } \]

Step 1 ：Rewrite the equation with the same base: \((2^2)^{-x} = (2^4)^{x+12}\)

Step 2 ：Simplify the equation using the power of a power rule: \(2^{-2x} = 2^{4x+48}\)

Step 3 ：Set the exponents equal to each other since the bases are the same: \(-2x = 4x + 48\)

Step 4 ：Solve for x: \(x = -8\)

Step 5 ：Final Answer: \(x = \boxed{-8}\)