Solve the following exponential equation without using logarithms. \[ 6^{-8 x}=216 \]

Step 1 ：Rewrite 216 as a power of 6. We know that \(6^3 = 216\), so we can rewrite the equation as \(6^{-8x} = 6^3\).

Step 2 ：Since the bases are the same, we can equate the exponents. So, \(-8x = 3\).

Step 3 ：Solve for x by dividing both sides by -8. \[x = \frac{3}{-8} = -\frac{3}{8}\]

Step 4 ：Check the solution. Substitute \(x = -\frac{3}{8}\) into the original equation: \[6^{-8(-\frac{3}{8})} = 6^3 = 216\]

Step 5 ：So, the solution to the equation \(6^{-8x} = 216\) is \(\boxed{x = -\frac{3}{8}}\)