Ask Me Help Desk - Find (√2+√3) - (√2-√3) as a single surd

Very interesting - every line of your work is incorrect, and yet you stumbled into the correct answer!

Starting with:

$(\sqrt 2 + \sqrt 3) - (\sqrt 2 - \sqrt 3)$

Eliminate the parentheses, being careful to watch signs:

$\sqrt 2 + \sqrt 3 - \sqrt 2 + \sqrt 3 = 2 \sqrt 3$

So like I said, you ended up with the correct answer. Now here's why your approach is all wrong:

1. First, $(\sqrt 2 + \sqrt 3) - (\sqrt 2 - \sqrt 3)$ does not equal $(\sqrt 2 + \sqrt 3)^2 - (\sqrt 2 - \sqrt 3)^2$ . This is like saying that $a +b = a^2 + b^2$ .

2. Next, $(\sqrt 2 + \sqrt 3)^2 - (\sqrt 2 - \sqrt 3)^2 = (\sqrt 2 )^2 + 2 \sqrt 2 \sqrt 3 + (\sqrt 3)^2 - [(\sqrt 2)^2 - 2 \sqrt2 \sqrt 3 + (\sqrt 3 )^2] = 2 + 2 \sqrt2 \sqrt 3 + 3 - (2 - 2 \sqrt 2 \sqrt 3 + 3) = 4 \sqrt 2 \sqrt 3$

3. Your third line is equivalent to zero: it's equivalent to a + b - a - b = 0

So I have no idea how you got the correct final answer, unless perhaps you copied it from an answer key?

Post back if you have additional questions about this.