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

1 Answer
Nov 20, 2016

Answer:

#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)#