1. **State the problem:** We have a radioactive element with an initial amount of 800 grams. The amount remaining after $t$ years is given by the formula: $$A = 800 \left(\frac{1}{64}\right)^{\frac{t}{300}}$$ We need to find how many grams remain after 200 years. 2. **Simplify the exponent:** The exponent is $\frac{t}{300}$. For $t=200$, this becomes: $$\frac{200}{300} = \frac{2}{3}$$ 3. **Rewrite the expression:** Substitute $t=200$: $$A = 800 \left(\frac{1}{64}\right)^{\frac{2}{3}}$$ 4. **Simplify the base:** Note that $64 = 4^3$, so: $$\left(\frac{1}{64}\right)^{\frac{2}{3}} = \left(64^{-1}\right)^{\frac{2}{3}} = 64^{-\frac{2}{3}} = (4^3)^{-\frac{2}{3}} = 4^{-2}$$ 5. **Evaluate the power:** $$4^{-2} = \frac{1}{4^2} = \frac{1}{16}$$ 6. **Calculate the remaining amount:** $$A = 800 \times \frac{1}{16} = 50$$ **Final answer:** After 200 years, 50 grams of the element remain. This corresponds to choice A.