@The-Blade-Dancer Thank you for asking and thanks for being curious.

The reason choice i.) is incorrect is because in general, for any numbers $$a$$ and $$b,$$ the following isn't true:

$$\sqrt{ a + b } = \sqrt{a} + \sqrt{b}$$

You could try some values of $$a$$ and $$b$$ to see this. Like for example, $$a = 1$$ and $$b = 4.$$ Then let's check our statement.

$$\sqrt{ 1 + 4 } \stackrel{?}{=} \sqrt{1} + \sqrt{4}$$

$$\sqrt{ 5} \stackrel{?}{=} 1 + 2$$
$$\sqrt{5} \neq 3$$

The answer choice i.) is claiming that the following is true:

$$\sqrt{ 6 + 5\sqrt{2}} = \sqrt{6} + \sqrt{5\sqrt{2}}$$

However, this is just the same as the statement $$\sqrt{a + b} = \sqrt{a} + \sqrt{b}$$ where $$a = 6$$ and $$b = 5\sqrt{2}.$$ And since we just showed that this statement is false, then choice i.) can't be true.