Factor $200=10^{2}\times 2$. Rewrite the square root of the product $\sqrt{10^{2}\times 2}$ as the product of square roots $\sqrt{10^{2}}\sqrt{2}$. Take the square root of $10^{2}$.
$$3\sqrt{2}-5\times 10\sqrt{2}+30\sqrt{8}$$
Multiply $-5$ and $10$ to get $-50$.
$$3\sqrt{2}-50\sqrt{2}+30\sqrt{8}$$
Combine $3\sqrt{2}$ and $-50\sqrt{2}$ to get $-47\sqrt{2}$.
$$-47\sqrt{2}+30\sqrt{8}$$
Factor $8=2^{2}\times 2$. Rewrite the square root of the product $\sqrt{2^{2}\times 2}$ as the product of square roots $\sqrt{2^{2}}\sqrt{2}$. Take the square root of $2^{2}$.
$$-47\sqrt{2}+30\times 2\sqrt{2}$$
Multiply $30$ and $2$ to get $60$.
$$-47\sqrt{2}+60\sqrt{2}$$
Combine $-47\sqrt{2}$ and $60\sqrt{2}$ to get $13\sqrt{2}$.
