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I wonder why Mathematica does not act on the powers inside square roots. See the following example:

Sqrt[a^4 b^4 c^4 d^4] does not give

a^2 b^2 c^2 d^2]

rather it returns back

Sqrt[a^4 b^4 c^4 d^4]

as the output.

Can someone comment on this?

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2 Answers 2

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Simplify[Sqrt[a^4  b^4  c^4  d^4]]

Gives

Sqrt[a^4 b^4 c^4 d^4]

Now

Assuming[Element[{a, b, c, d}, Reals], Simplify[Sqrt[a^4  b^4  c^4  d^4]]]

Gives

a^2 b^2 c^2 d^2

Mathematica works in complex domain by default. A simple example

a = 6*I;

Now

Sqrt[a^4]
(* 36 *)

But

a^2
(* -36 *)
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I think PowerExpand is what you need as in

PowerExpand[Sqrt[a^4  b^4  c^4  d^4]]
(* a^2*b^2*c^2*d^2 *)
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