Question

How do you simplify `sqrt (16) / (sqrt (4) + sqrt (2))`?

Answer 1

It is

#sqrt (16) / (sqrt (4) + sqrt (2))=4/[sqrt2(sqrt2+1)]= 2sqrt2/(sqrt2+1)=2*sqrt2(sqrt2-1)/[(sqrt2+1)*(sqrt2-1)]= 2sqrt2(sqrt2-1)#

Answer 2

`4 - 2 sqrt 2`

Explanation:

Try to rationalize the denominator. Multiply numerator and denominatr by `(sqrt 4 - sqrt 2)`

`sqrt 16 ( sqrt 4 - sqrt 2) / ( (sqrt 4+ sqrt 2 ) * (sqrt 4 - sqrt 2 ) )`

`4 * (2 - sqrt 2) / ( 4 - 2)`

`4 * (2 - sqrt 2) / 2`

`2 * (2 - sqrt 2)`

`4 - 2sqrt2`

Answer 3

Multiply through by `(2-sqrt2)/(2-sqrt2)` and work through to get `4-2sqrt2=2(2-sqrt2)`

Explanation:

Let's start with the original:

`sqrt16/(sqrt4+sqrt2)`

Let's first take the square roots of the perfect squares:

`4/(2+sqrt2)`

In order to simplify, we need the square root out from the denominator. The way to do this is to ensure that when we do FOIL (the process of multiplying 2 quantities within brackets), we don't end up with more square roots. To do that, we'll multiply by `(2-sqrt2)` which will eliminate that possibility (like this):

`4/(2+sqrt2)*((2-sqrt2)/(2-sqrt2))`

`(4*2-4sqrt2)/(2*2-2sqrt2+2sqrt2-sqrt2sqrt2)`

`(8-4sqrt2)/(4-2)=(8-4sqrt2)/2=4-2sqrt2=2(2-sqrt2)`