You're on a roll — Step 1: Solve $12\sqrt{17} - 8\sqrt{17}$. These are like surds, meaning they have the same radical part ($\sqrt{17}$). To subtract them, subtract their coefficients. $$12\sqrt{17} - 8\sqrt{17} = (12 - 8)\sqrt{17}$$ $$= 4\sqrt{17}$$ $$\boxed{4\sqrt{17}}$$ Step 2: Solve $5\sqrt{5} + \sqrt{5}$. These are like surds. To add them, add their coefficients. Remember that $\sqrt{5}$ is equivalent to $1\sqrt{5}$. $$5\sqrt{5} + 1\sqrt{5} = (5 + 1)\sqrt{5}$$ $$= 6\sqrt{5}$$ $$\boxed{6\sqrt{5}}$$ Step 3: Solve $5\sqrt{7} \times \frac{2}{5}\sqrt{7}$. Multiply the coefficients together and the surd parts together. $$(5 \times \frac{2}{5}) \times (\sqrt{7} \times \sqrt{7})$$ First, multiply the coefficients: $$5 \times \frac{2}{5} = \frac{10}{5} = 2$$ Next, multiply the surd parts: $$\sqrt{7} \times \sqrt{7} = \sqrt{7 \times 7} = \sqrt{49} = 7$$ Finally, multiply these two results: $$2 \times 7 = 14$$ $$\boxed{14}$$ Step 4: Simplify by expressing $(3\sqrt{2})^2$ as a complete surd. To simplify $(3\sqrt{2})^2$, square both the coefficient and the surd part. $$(3\sqrt{2})^2 = 3^2 \times (\sqrt{2})^2$$ $$= 9 \times 2$$ $$= 18$$ To express 18 as a complete surd, write it under a square root sign. $$18 = \sqrt{18^2}$$ $$= \sqrt{324}$$ $$\boxed{\sqrt{324}}$$ What's next?