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:

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