Question

Help: Radical expression

Answer 1

\[ \frac{ (5-(1/\sqrt{5})) }{ 1+(1/\sqrt{5})) } \]

Answer 2

here's a better representation
\[\frac{ 5-\frac{ 1 }{ \sqrt{5} } }{ 1+\frac{ 1 }{ \sqrt{5} } }\]

Answer 3

\(\huge\color{black}{ \frac{5-\frac{1}{\sqrt{5}}}{1+\frac{1}{\sqrt{5}}} }\)


\(\huge\color{black}{ \frac{\frac{5\sqrt{5}}{\sqrt{5}}-\frac{1}{\sqrt{5}}}{\frac{\sqrt{5}}{\sqrt{5}}+\frac{1 }{\sqrt{5}}} }\)

Answer 4

\(\huge\color{black}{ \frac{\frac{5\sqrt{5}}{\sqrt{5}}-\frac{1}{\sqrt{5}}}{\frac{\sqrt{5}}{\sqrt{5}}+\frac{1 }{\sqrt{5}}} }\)


\(\huge\color{black}{ \frac{\frac{5\sqrt{5}-1}{\sqrt{5}}}{\frac{\sqrt{5}+1}{\sqrt{5}}} }\)

Answer 5

what i have so far:

numerator- \[\frac{ 5\sqrt{5}-1 }{ \sqrt{5} }\]
denominator- \[\frac{\sqrt{5}+1 }{ \sqrt{5} }\]

Answer 6

divide top and bottom by square root of 5.

Answer 7

and you are correct for what you got so far... now divide by square root of 5 on top and bottom, you get ?

Answer 8

so to do that i would take the reciprocal of the denominator and multiply it with the numerator?

Answer 9

yes, in other words, you flip the second fraction, and multiply

Answer 10

Good that you are familiar with the rules. That makes me smile.

Answer 11

\[\frac{ \sqrt{5}(5\sqrt{5}-1) }{ \sqrt{5}(\sqrt{5}+1) } = \frac{ 25 - \sqrt{5} }{ 5+\sqrt{5} }\]

Answer 12

i think i messed up while distributing

Answer 13

I am taking the first side of the last thing you wrote. And THERE, why can't you just cancel the square root of 5 on top and bottom ?

Answer 14

the square root of 5, on top and bottom , those that are in front of the parenthesis.

Answer 15

oh wow

Answer 16

And you didn't mess up. You are completely correct.

Answer 17

you're right

Answer 18

wow ?

Answer 19

i should've just cancelled the sqrt5

Answer 20

Yeah.. just go ahead and cancel the square roots of 5.

Answer 21

tell me what you get after that.

Answer 22

So the answer is 5sqrt5 - 1 over sqrt5 + 1

Answer 23

yes

Answer 24

but this is not the answer.

Answer 25

you are not allowed to have a radical in the denominator.
Multiply both sides times the denominator's conjugate.

Answer 26

oh okay. i need rationalize the denominator right?

Answer 27

yeaahh !!! :)

Answer 28

okay, i'll do that right now. give me a sec

Answer 29

sure :) I'll just retype the last step.

Answer 30

\(\huge\color{ blue }{\huge {\bbox[5pt, cyan ,border:2px solid purple ]{ \frac{5\sqrt{5}-1}{1+\sqrt{5}} }}}\)

Answer 31

thanks so much!

Answer 32

I'm almost done lol

Answer 33

yes, I would prefer you to finish the problem here, so that I can make sure it is correct. (Not like I am a teach.. but)

Answer 34

\[\frac{ (5\sqrt{5}-1)(\sqrt{5}-1) }{ (\sqrt{5)^2}-(1)^2 }= \frac{ ? }{ 4 }\]

Answer 35

okay so i have to distribute the numerator

Answer 36

yes, you would need to just
simplify" distribute, or whatever you want to call it ... the denominator.

First though, I would finish doing the numerator.

Answer 37

\[numerator- 25 - 5\sqrt{5} - \sqrt{5} + 1\]

Answer 38

you are multiplying by \(\normalsize\color{black}{ \sqrt{5}-1}\) on top and bottom, no ?

Answer 39

and that is no the same as \(\normalsize\color{black}{ 1-\sqrt{5}}\)

Answer 40

okay thanks. i corrected it

Answer 41

wait, but I'm still confused. what's the next step? I'm still stuck at 5sqrt5 - 1 / sqrt5 +1

Answer 42

\(\LARGE\color{black}{ \frac{5\sqrt{5}-1}{\sqrt{5}+1} }\)


\(\LARGE\color{black}{ \frac{(5\sqrt{5}-1)\color{red}{(\sqrt{5}-1)}}{(\sqrt{5}+1)\color{red}{(\sqrt{5}-1)}} }\)


\(\LARGE\color{black}{ \frac{(5\sqrt{5}-1)\color{red}{(\sqrt{5}-1)}}{\sqrt{5}^2-1^2} }\)


\(\LARGE\color{black}{ \frac{(5\sqrt{5}-1)\color{red}{(\sqrt{5}-1)}}{5-1} }\)


\(\LARGE\color{black}{ \frac{(5\sqrt{5}-1)\color{red}{(\sqrt{5}-1)}}{4} }\)

Answer 43

Tell me if you are god with THIS ...

Answer 44

okay, yeah I'm good with that. that's what i had before

Answer 45

Yes, but you then multiplied times the 1-sqrt5, instead of multiplying times sqrt5-1

Answer 46

oh okay, i see what you're saying now

Answer 47

Yes, and your answer is going to be ? ....

Answer 48

\[ \frac{ 25 - 6\sqrt{5} + 1 }{ 4 }\]

Answer 49

yes, and then add like terms on the top,
and divide by 2, (you can call it "war" or cancel 2) on top and bottom.

Answer 50

or \[\frac{ 26 - 6\sqrt{5} }{ 4}\]

Answer 51

okay so 13 - 6sqrt5 / 2

Answer 52

not sqrt6.

You divide the entire top by 2.

Answer 53

oh, so \[\frac{ 13 - 3\sqrt{5} }{ 2 }\]

Answer 54

yup !

Answer 55

okay thank you very much!!

Answer 56

You welcome !