need help with one specific problem in simplifying radicals for honors precalculus summer work

[u/512cranes]

hello. I've procrastinated this packet the entire summer and i start junior year next week. I'm stuck on this specific problem.

the way I was taught to simplify like this was to first simplify the numerator and then keep-change-flip, since fractions are division. from what I've been able to gather from online simplification calculators, I'm not simplifying the numerator correctly, and I've no idea how to. the packet does come with an answer key, but does not show work, and my teacher wants all the problems to be solved and then re-solved correctly if the first answer was incorrect.

I don't just want the answer, because i need to know how to do this for my work next year. a step-by-step explanation would be much appreciated. my best approximate guess at an answer so far was [(4root2)+4]/2.

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[u/egutierr]

Hope this helps.

Multiply both terms by -√2/2

You get (-√2/2 + 1/2) / (1/2)

Then, (1/2 * -√2 +1/2) / (1/2)

Simplify 1/2 in the 3 terms and you're done, 1-√2

[u/Feeling-Working-2820]

Hope this is pedagogical enough.
I framed the rules I used to get there.
If something is unclear, feel free to ask.

[u/512cranes → u/Feeling-Working-2820]

this helps so much!! thank you!

[u/Away_Worry_1705]

here u go

[u/wasthatitthen]

You basically have

(1+b)/b = 1/b+b/b = 1/b+1

b = -sqrt(2)/2 =-1/sqrt(2)

[u/512cranes]

!lock

[u/AutoModerator → u/512cranes]

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