Question

Simplify
\({\sqrt(2)-\sqrt(6)}/{\sqrt(2)+\sqrt(6)}\)

Answer 1

\[\huge\color{blue}{ \frac{ \sqrt{2} - \sqrt{6} }{ \sqrt{2} + \sqrt{6} } }\]

Answer 2

Multiply top and bottom by conjugate, or by \[\sqrt{2} - \sqrt{6}\]

Answer 3

\[\huge\color{red}{ \frac{ \sqrt{2} - \sqrt{6} }{ \sqrt{2} + \sqrt{6} } } \times \huge\color{red}{ \frac{ \sqrt{2} - \sqrt{6} }{ \sqrt{2} - \sqrt{6} } }\]

Answer 4

Ok i see that but how to I multiple them together?

Answer 5

I never multiplied squareroots

Answer 6

\[\huge\color{green}{ \frac{ (\sqrt{2} - \sqrt{6})( \sqrt{2} - \sqrt{6}) }{ (\sqrt{2} + \sqrt{6} ) (\sqrt{2}- \sqrt{6})} }\]

Answer 7

\[\huge\color{red}{ \sqrt{x} \times \sqrt{x} =x }\]

Answer 8

On the bottom, the square root are going to cancel.

Answer 9

oh so
would the top be 2 -6?

Answer 10

Close, on the top there is also a middle term.

Answer 11

\[\huge\color{red}{ (a-b)^2=a^2-2ab+b^2 }\]use this formula to do the top.

Answer 12

\[(2 - 6)^2 = 2^2 - 2(2)(-6)+(-6)^2\]

Answer 13

64?

Answer 14

Well, it's \[( \sqrt {2}-\sqrt {6} )^2\]

Answer 15

And not \[(2-6)^2\]

Answer 16

oh how would we do it with the squareroots?

Answer 17

\[(\sqrt{2}-\sqrt{6})^2=\sqrt{2}~^2~~-~~2\times \sqrt{2} \times \sqrt{6}~~-~~(-\sqrt{6})~^2\]

Answer 18

\[2-2\sqrt{12}+6\]\[8-2\sqrt{12}\]\[8-4\sqrt{3}\]

Answer 19

ok i get that now but how would we do the bottom you said it cancels out

Answer 20

Sorry, I got disconnected.

Answer 21

\[(a-b)(a+b)=a^2-b^2\]\[(\sqrt{2}-\sqrt{6})~(\sqrt{2}+\sqrt{6})~=~\sqrt{2}~^2~~~-~~~\sqrt{6}~^2=~2~-~6~~=~~-4\]

Answer 22

\(-2\sqrt{3}\)?

Answer 23

No, it's not, \[-2\sqrt{3}\]

Answer 24

Do you see how I did the bottom?

Answer 25

oh wait so you got -4 for the bottom?
is see that somewhat

Answer 26

Yes, -4 on the bottom.

Answer 27

So so far we have \[\huge\color{blue}{ \frac{8 - 4\sqrt{3}}{-4} }\]

Answer 28

Right?

Answer 29

yes

Answer 30

8-4=2
so \(2\sqrt3\) the top

Answer 31

You can't subtract those, they are not like terms.

One is \[8~~~~~(whole~~number)\]

and the other is \[4\sqrt{3}~~~~~~~~(radical)\]

Answer 32

oh so how would I simplify those?

Answer 33

factor the top out of -4.
\[\huge\color{blue}{ \frac{8 - 4\sqrt{3}}{-4} }\]
\[\huge\color{blue}{ \frac{-4(-2 +1\sqrt{3})}{-4} }\]
\[\huge\color{blue}{ \frac{-4(-2 +\sqrt{3})}{-4} }\]
\[\huge\color{blue}{ (-2 +\sqrt{3}) }\]

Answer 34

how did you factor out 8-4sqrt(3)
-4*-2 = 8
-4*1 = -4
oh never mind got it thank you so much :D

Answer 35

Are you sure you got it?

Answer 36

yup the -4 cancels the -4 on the bottom leaving us with -2srt(3)

Answer 37

well, not exactly, it's \[-2 + \sqrt{3}\]we didn't add, just factored out of -4.

Answer 38

oh okay I understand now :) thanks

Answer 39

YW!