How do you simplify 2/(sqrt(1+sqrt(2))-1) ?

1 Answer
George C.
Nov 20, 2016

2/(sqrt(1+sqrt(2))-1) = sqrt(2+2sqrt(2))+sqrt(2)

Explanation:

The difference of squares identity can be written:

a^2-b^2 = (a-b)(a+b)

We find:

2/(sqrt(1+sqrt(2))-1) = (2(sqrt(1+sqrt(2))+1))/((sqrt(1+sqrt(2))-1)(sqrt(1+sqrt(2))+1))

color(white)(2/(sqrt(1+sqrt(2))-1)) = (2sqrt(1+sqrt(2))+2)/((color(red)(cancel(color(black)(1)))+sqrt(2))-color(red)(cancel(color(black)(1))))

color(white)(2/(sqrt(1+sqrt(2))-1)) = (2sqrt(1+sqrt(2))+2)/(sqrt(2))

color(white)(2/(sqrt(1+sqrt(2))-1)) = sqrt(2)sqrt(1+sqrt(2))+sqrt(2)

color(white)(2/(sqrt(1+sqrt(2))-1)) = sqrt(2+2sqrt(2))+sqrt(2)

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