# Simplify product of square roots

What assumption is needed to make both outputs equal to Sqrt[P^2]/2

In[65]:= tps = (1/2) (1 - T (1 - P^2) + Sqrt[(1 - T(1 - P^2))^2 - P^2]);
tms = (1/2) (1 - T (1 - P^2) - Sqrt[(1 - T(1 - P^2))^2 - P^2]);
FullSimplify[Sqrt[tms tps]]
FullSimplify[Sqrt[tms] Sqrt[tps]]

Out[67]= Sqrt[P^2]/2

Out[68]= 1/2 Sqrt[1 + (-1 + P^2) T - Sqrt[(-1 + P^2) (-1 + T (2 + (-1 + P^2) T))]]
Sqrt[1 + (-1 + P^2) T + Sqrt[(-1 + P^2) (-1 + T (2 + (-1 + P^2) T))]]


Comment: There are many similar questions on the web but I couldn't find a proper answer. To assume that T and P are real isn't enough.

Edit: T and P are real numbers and not necessarily equal to zero. So the question should better be, what is the reason that both ways don't yield the same result and how can I change this?

• Any bounds at all on either T or P? – Michael E2 Sep 3 '14 at 23:11
• Instead of assumptions, you could just force it by using a replacement rule (only recommended if you can verify the validity in the end): Simplify@ReleaseHold[Hold[Sqrt[tms] Sqrt[tps]] //. HoldPattern[Sqrt[x_] Sqrt[y_]] :> Sqrt[x y]] – Jens Sep 3 '14 at 23:22

In Mathematica, Sqrt[x]Sqrt[y]==Sqrt[x y] is not always true, there is a simple example gived by @Rahul Narain

$\sqrt{-1}\times\sqrt{-1}=i\times i=-1$

$\sqrt{(-1)\times(-1)}=\sqrt{1}=1$

So FullSimplify[Sqrt[x] Sqrt[y]] don't give the result as Sqrt[x y]

To get a better answer, to assume range of the variables is helpful, like this

In[59]:= tps = (1/2) (1 - T (1 - P^2) + Sqrt[(1 - T (1 - P^2))^2 - P^2]);
tms = (1/2) (1 - T (1 - P^2) - Sqrt[(1 - T (1 - P^2))^2 - P^2]);
FullSimplify[Sqrt[tms tps]]
FullSimplify[Sqrt[tms] Sqrt[tps], Assumptions -> tps >= 0 && tms >= 0]

Out[61]= Sqrt[P^2]/2

Out[62]= Abs[P]/2

• It is not always true that $\sqrt x\sqrt y=\sqrt{xy}$ when $x$ and $y$ are real; for example, take $x=y=-1$. If $x$ and $y$ are positive then it holds. – Rahul Sep 4 '14 at 3:46
• Oh I understand, thanks! – wuyingddg Sep 4 '14 at 4:07
• @Rahul Narain, thanks a lot. How could I forget this!? – PeMa Sep 4 '14 at 8:17