Solve 12sqrt(17) - 8sqrt(17).

These are like surds, meaning they have the same radical part ($\sqrt{17}$). To subtract them, subtract their coefficients.

$$12\sqrt{17} - 8\sqrt{17} = (12 - 8)\sqrt{17}$$ $$= 4\sqrt{17}$$ $$\boxed{4\sqrt{17}}$$

These are like surds. To add them, add their coefficients. Remember that $\sqrt{5}$ is equivalent to $1\sqrt{5}$.

$$5\sqrt{5} + 1\sqrt{5} = (5 + 1)\sqrt{5}$$ $$= 6\sqrt{5}$$ $$\boxed{6\sqrt{5}}$$

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:

$$2 \times 7 = 14$$ $$\boxed{14}$$

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.

$$18 = \sqrt{18^2}$$ $$= \sqrt{324}$$ $$\boxed{\sqrt{324}}$$