Question

Tutor question #3
Simplify:
\[\sqrt{5-\sqrt{21}}\]
(conditions: pen, paper, no calculator, done in 2 minutes)
*

Answer 1

what if x = \(\sqrt{5-\sqrt{21}}\)
then \(x^2 = 5 - \sqrt{21}\)

\[x^2 - 5 = -\sqrt{21}\]
\[(x^2 - 5)^2 = 21\]

oh goodness quadratic o.O

Answer 2

actually it is a 4th degree equation

Answer 3

\[(x^2-5)^2=21\]\[x^4-10x^2+25=21\]\[x^4-10x^2+4=0\]\[x^4-10x^2+4=0\]\[x^2=\frac{10\pm \sqrt{(-10)^6-4(4)}}{2}\]\[x^2=5\pm \sqrt{21}\]Sorry... are you sure that it is 'simplified'?

Answer 4

Oh.that's power 2, typo

Answer 5

apparently it is also
\[\sqrt{\frac{7}{2}}-\sqrt{\frac{3}{2}}\] i remember seeing something like this before but i am not sure i remember how to go from one to the other

Answer 6

I need to know how to work that out.....

Answer 7

cant seem to get it, i think it was a trick

Answer 8

............................
It... was ... a ... question ... asked ... when ... my friend applied for a summer job ...

Answer 9

really? doing what??

Answer 10

tutor - teaching high school students

Answer 11

hmm i wonder what answer they wanted

Answer 12

Same here...

Answer 13

When I was doing some exercises few days ago, I saw similar questions, but clearer, like this:
express \(\sqrt{28-2\sqrt{147}}\) in the form of \(\sqrt{x}-\sqrt{y}\). I can still handle this. But that one, I failed :(

Answer 14

multiplying by the conjugate give \(\frac{2}{\sqrt{5+\sqrt{21}}}\) think

Answer 15

any examples anywhere?

Answer 16

example?!

Answer 17

You want to show \[\sqrt{5-\sqrt{21}} \text{ equals } \frac{\sqrt{7}-\sqrt{3}}{2} ?\]

was just wondering if you have an example for writing that one thing in that other form

Answer 18

oops sqrt(2) on bottom

Answer 19

how did you get that identity?

Answer 20

From experience...

Answer 21

lol great experience
I want to prove that identity lol

Answer 22

First, \((\sqrt{a} - \sqrt{b})^2\) = a + b -2\sqrt{ab}
For a>b
\[\sqrt{5-\sqrt{21}} = \sqrt{5-2\sqrt{\frac{21}{4}}}\]

Now,
a+b = 5 => a=5-b
ab = 21/4
(5-b)b = 21/4
-4b^2 + 20b - 21 =0
b=1.5 or b =3.5 (rejected)
a = 5 - 1.5 = 3.5
So, it is \(\sqrt{\frac{7}{2}}-\sqrt{\frac{3}{2}}\)
Does that make sense?

Answer 23

ok i see that identity :)

Answer 24

that one is easy to prove

Answer 25

maybe because it is an actual identity, right? :p

Answer 26

Perfect square is perfect :)

Answer 27

Does that make sense?
Apart from the latex fail...

Answer 28

very interesting
i wouldn't have thought of that

Answer 29

I'm going to post the link to satellite73's post then. He doesn't even come to check..

Answer 30

Great work @Callisto :)

Answer 31

And thank you for all your time!!!!