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

3 Answers
Konstantinos Michailidis
Jun 26, 2016

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)√16√4+√2=4√2(√2+1)=2√2√2+1=2⋅√2√2−1(√2+1)⋅(√2−1)=2√2(√2−1)

jags
Jun 26, 2016

4 - 2 sqrt 24−2√2

Explanation:

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

sqrt 16 ( sqrt 4 - sqrt 2) / ( (sqrt 4+ sqrt 2 ) * (sqrt 4 - sqrt 2 ) )√16√4−√2(√4+√2)⋅(√4−√2)

4 * (2 - sqrt 2) / ( 4 - 2) 4⋅2−√24−2

4 * (2 - sqrt 2) / 2 4⋅2−√22

2 * (2 - sqrt 2) 2⋅(2−√2)

4 - 2sqrt24−2√2

MathFact-orials.blogspot.com
Jun 26, 2016

Multiply through by (2-sqrt2)/(2-sqrt2)2−√22−√2 and work through to get 4-2sqrt2=2(2-sqrt2)4−2√2=2(2−√2)

Explanation:

Let's start with the original:

sqrt16/(sqrt4+sqrt2)√16√4+√2

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

4/(2+sqrt2)42+√2

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)(2−√2) which will eliminate that possibility (like this):

4/(2+sqrt2)*((2-sqrt2)/(2-sqrt2))42+√2⋅(2−√22−√2)

(4*2-4sqrt2)/(2*2-2sqrt2+2sqrt2-sqrt2sqrt2)4⋅2−4√22⋅2−2√2+2√2−√2√2

(8-4sqrt2)/(4-2)=(8-4sqrt2)/2=4-2sqrt2=2(2-sqrt2)8−4√24−2=8−4√22=4−2√2=2(2−√2)

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