1. The problem is to calculate $8 \times 6 \frac{1}{4}$ using the distributive law. 2. First, convert the mixed number $6 \frac{1}{4}$ to an improper fraction: $$6 \frac{1}{4} = \frac{6 \times 4 + 1}{4} = \frac{24 + 1}{4} = \frac{25}{4}.$$ 3. Now rewrite the multiplication: $$8 \times \frac{25}{4}.$$ 4. Use the distributive law by expressing 8 as $\frac{8}{1}$ and multiply numerators and denominators: $$\frac{8}{1} \times \frac{25}{4} = \frac{8 \times 25}{1 \times 4} = \frac{200}{4}.$$ 5. Simplify the fraction: $$\frac{200}{4} = 50.$$ 6. Therefore, the final answer is $50$.