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QUESTION

When I try and simplify this expression:

In[1]:= FullSimplify[Sqrt[ x^2 + 2 y^2 + 2 y Sqrt[x^2 + y^2] ], (x | y) \[Element] Reals]
Out[1]= Sqrt[x^2 + 2 y (y + Sqrt[x^2 + y^2])]

it does not simplify properly.
However, Mathematica knows it equals a simpler expression:

In[2]:= FullSimplify[ Sqrt[x^2 + 2 y^2 + 2 y Sqrt[x^2 + y^2] ] == Sqrt[x^2 + y^2] + y, (x | y) \[Element] Reals]
Out[2]= True

Why doesn't Mathematica simplify this expression, and how can I make it do it anyway, even if the expression is part of a larger expression?

{ asked by Joe }

ANSWER

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 ...

{ answered by TheDoctor }
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