1. Calculate $$4\sqrt{17} \cdot 5\sqrt{2} \cdot \sqrt{34}$$.

Let's break this down step-by-step:

1. We are asked to calculate the product of three terms: $$4\sqrt{17}$$, $$5\sqrt{2}$$, and $$\sqrt{34}$$.
2. First, we can multiply the coefficients (the numbers outside the square roots): $$4 \times 5 = 20$$.
3. Next, we multiply the numbers inside the square roots: $$\sqrt{17} \times \sqrt{2} \times \sqrt{34}$$.
4. We can combine these under a single square root: $$\sqrt{17 \times 2 \times 34}$$.
5. Calculate the product inside the square root: $$17 \times 2 = 34$$, and $$34 \times 34 = 34^2$$.
6. So, we have $$\sqrt{34^2}$$.
7. The square root of $$34^2$$ is 34.
8. Now, combine the coefficient and the result from the square root: $$20 \times 34$$.
9. $$20 \times 34 = 680$$.

Ответ: 680