Solution: Set $ \frac{1}{8}D_0 = D_0 \cdot 2^{-t/12} $. Divide both sides by $ D_0 $: $ \frac{1}{8} = 2^{-t/12} $. Note $ \frac{1}{8} = 2^{-3} $, so $ 2^{-3} = 2^{-t/12} $. Equate exponents: $ -3 = -\frac{t}{12} $. Solve for $ t $: $ t = 36 $.

["Title: Solving Exponential Decay: How to Determine the Half-Life from Decay Formula", "---", "Understanding radioactive decay or exponential decay models is essential in physics, chemistry, and engineering. One powerful method involves solving equations of the form $ \frac{1}{8}D_0 = D_0 \cdot 2^{-t/12} $, a common form used to model decay over time. Here’s a clear step-by-step solution explaining how to solve this equation and find the decay time $ t $.", "---", "### The Decay Equation", "We begin with the exponential decay equation:", "$$\n\frac{1}{8}D_0 = D_0 \cdot 2^{-t/12}\n$$", "Step 1: Divide both sides by $ D_0 $", "Since $ D_0 $ represents an initial quantity and is nonzero, we can safely divide both sides by $ D_0 $:", "$$\n\frac{1}{8} = 2^{-t/12}\n$$", "---", "### Rewriting $ \frac{1}{8} $ as a Power of 2", "We recognize that $ \frac{1}{8} = 2^{-3} $, because:", "$$\n2^{-3} = \frac{1}{2^3} = \frac{1}{8}\n$$", "Substituting this into the equation gives:", "$$\n2^{-3} = 2^{-t/12}\n$$", "---", "### Equating the Exponents", "Because the bases are equal (both base 2), we can equate the exponents:", "$$\n-3 = -\frac{t}{12}\n$$", "---", "### Solving for $ t $", "Multiply both sides by $-12$ to isolate $ t $:", "$$\nt = 3 \ imes 12 = 36\n$$", "---", "### Conclusion", "The solution to $ \frac{1}{8}D_0 = D_0 \cdot 2^{-t/12} $ is $ t = 36 $. This means the quantity reduced to one-eighth of its initial value after 36 time units (e.g., half-lives) in a decay process following an exponential model with a half-life of 12 units.", "---", "This mathematical approach applies directly to real-world scenarios like carbon dating, nuclear decay, and chemical reactions governed by exponential decay laws. Mastering such equations strengthens your ability to analyze decay processes accurately.", "---", "Key takeaways:\n- Always divide by the initial quantity to isolate the decay term.\n- Recognize powers of 2 for faster simplification.\n- Equating exponents is valid when bases are equal.\n- Applying algebra to decay equations enhances problem-solving in science and engineering.", "---", "For further exploration: See how exponential models apply to half-life calculations in radioactive isotopes and environmental science."]