Why doesn't Mathematica simplify a square root of an expression that equals a square of a positive real?
Change to polar coordinates, {x -> r Cos[t], y -> r Sin[t]}:
FullSimplify[ Sqrt[2 y Sqrt[x^2 + y^2] + x^2 + 2y^2] /.
{x -> r Cos[t], y -> r Sin[t]}, r > 0 && 0 < t< 2Pi]
r (Sin[t] + 1)
This should simplify your long expression ...
Possibly because the second simplification request is much easier than the first. Consider:
Also, the second FullSimplify
can be done by Simplify
. Not surprising since all MMA has to do is square both sides.
edit: Perhaps this will suggest an approach:
You might do this:
Simplify[Sqrt[x^2 + 2 y^2 + 2 y Sqrt[x^2 + y^2]] /. x^2 -> u^2 - y^2, {u > 0, y > 0}] /. u -> Sqrt[x^2 + y^2]
The result is here: y + Sqrt[x^2 + y^2]
I required that both u
and y
are positive. If they are not:
Simplify[Sqrt[x^2 + 2 y^2 + 2 y Sqrt[x^2 + y^2]] /.x^2 -> u^2 - y^2, {u < 0, y < 0}] /. u -> Sqrt[x^2 + y^2]
You get another result: Abs[-y + Sqrt[x^2 + y^2]]