What is the square root of 2 plus the square root of 3?

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Answer 1 · 2015-05-29T20:16:34

#sqrt(2)+sqrt(3)# is not readily simplifiable.

You can calculate an approximate value as:

#sqrt(2)+sqrt(3) ~= 1.414213562 + 1.732050808 = 3.146264370#

Funnily enough, I was intrigued the other day to find the simplest polynomial with integer coefficients of which #sqrt(2)+sqrt(3)# is a root.

The answer is: #x^4-10x^2+1 = 0#

which has roots:

#sqrt(2)+sqrt(3)#
#sqrt(2)-sqrt(3)#
#-sqrt(2)+sqrt(3)#
#-sqrt(2)-sqrt(3)#

and #x^4-10x^2+1# has factors:

#(x-sqrt(2)-sqrt(3))#
#(x-sqrt(2)+sqrt(3))#
#(x+sqrt(2)-sqrt(3))#
#(x+sqrt(2)+sqrt(3))#