Question

(√(2) + √(6)) / √(2 + √(3))

The calculated answer is 2. How would I prove it without using a calculator?

Answer 1

look how sqrt(3) can also be sqrt(300/100), then break that up to get sqrt(300)/sqrt(100) which will be sqrt(300)/10. So, from there, notice that 17^2 = 289 and 18^2 = 324. So, 17.5 falls between the two. So, sqrt(300) = approximately 17.5. Still keep in mind that the denominator is 10. So you have 17.5 / 10 which is 1.75.

Follow this pattern for the rest of them. I hope this makes sense (:

Answer 2

\[\sqrt{2} + \sqrt{3}\]

Is that right?

Answer 3

\[\sqrt{2+\sqrt{3}}\]

Or is it this?

Answer 4

Sorry for the confusion. It's the second one.

Answer 5

Top like that as well?

Answer 6

Let me just rewrite that :)

\[(\sqrt{2} + \sqrt{6}) / \sqrt{2 + \sqrt{3}}\]

Answer 7

k, let me look...

Answer 8

Well, I haven't done it but there is a standard sort of approach to these things..

U want to get rid of radicals on the bottom where u can so start off by multiplying top and bottom by the bottom to get rid of the first (outer) square root

Answer 9

Then you have 2 + sqrt 3 only on the bottom and to get rid of that square root multiply by 2 - sqrt 3.

Answer 10

That brings the bottom to 1.

Answer 11

Sorry, busy...

With me so far...?

Answer 12

You should just have a numerator now composed of three pieces including the ugly one, right?

Answer 13

Anybody there?

Answer 14

I'm still here :) Just trying to make sense of the numerator...

Answer 15

Now I think we should square it so we can deal with the ugly bit and we will square root it again after...

Answer 16

Squaring the 1 on the bottom doesn't change it.

Answer 17

U need to multiply it all out...

Answer 18

And simplify.

Answer 19

I'm sorry; you lost me there. so the numerator is made up of \[(\sqrt{2} + \sqrt{6}) (\sqrt{2 + \sqrt{3}}) ( 2 - \sqrt{3)}\] right? and I multiply it out and simplify.

Answer 20

U can't multiply it like that because of the ugly bit so you have to square the whole thing in order to simplify it and then after it is simplified remember to take the aqrt to get back to where u started.

Obviously we want it to come out to 4 and guess what, it does:-)

Answer 21

What a mess, it was easier with a calculator.....

Answer 22

http://www.wolframalpha.com/input/?i=%28sqrt2%2Bsqrt6%29*sqrt%282%2Bsqrt3%29*%282-sqrt3%29

Answer 23

U get it yet? Page full of multiplying out and simplifying....

Answer 24

Still trying to work it out...

Answer 25

Anyway, you know how to do these in the future, right?

Answer 26

Yes, thanks for the help :)

Answer 27

ur welcome;-)

Answer 28

It is much less work if you just square the thing first rather than later:-(

Then it just drops out easily in a couple lines, I should have looked at it abit more closely before jumping in, my apologies.

Still, good practice with radicals:-)

Answer 29

haha ok thanks :)