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

3 Answers

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)

Jun 26, 2016

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

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)