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

May 29, 2015

$\sqrt{2} + \sqrt{3}$ is not readily simplifiable.

You can calculate an approximate value as:

$\sqrt{2} + \sqrt{3} \cong 1.414213562 + 1.732050808 = 3.146264370$

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: ${x}^{4} - 10 {x}^{2} + 1 = 0$

which has roots:

$\sqrt{2} + \sqrt{3}$
$\sqrt{2} - \sqrt{3}$
$- \sqrt{2} + \sqrt{3}$
$- \sqrt{2} - \sqrt{3}$

and ${x}^{4} - 10 {x}^{2} + 1$ has factors:

$\left(x - \sqrt{2} - \sqrt{3}\right)$
$\left(x - \sqrt{2} + \sqrt{3}\right)$
$\left(x + \sqrt{2} - \sqrt{3}\right)$
$\left(x + \sqrt{2} + \sqrt{3}\right)$