Why does choice i. in the mini question not work? I think it's fine.

]]>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.

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