Two equivalent expressions give False with ===

Questions
1

Could someone explain why these two expressions are not equivalent?

Sqrt[(a^2 + b^2)^7] === (a^2 + b^2)^(7/2)

Thanks.

2

=== checks for equality of expressions. Compare

FullForm[Sqrt[(a^2 + b^2)^7]]

Power[Power[Plus[Power[a, 2], Power[b, 2]], 7], Rational[1, 2]]

with

FullForm[(a^2 + b^2)^(7/2)]

Power[Plus[Power[a, 2], Power[b, 2]], Rational[7, 2]]

3

Okay. Agreed. The way Mathematica parses these expressions is different. But algebraically these two expressions are equivalent. Is there anyway to expand them to demonstrate this fact? I’ve tried

Sqrt[(a^2+b^2)^7//Expand] and Expand[(a^2+b^2)^(7/2)] with no luck. Thoughts? (And thanks).

(By simply checking (3^2+4^2)^(7/2) and Sqrt[(3^2+4^2)]^7 I, of course, get the same answer)

4

PowerExpand[Sqrt[(a^2 + b^2)^7]] === PowerExpand[(a^2 + b^2)^(7/2)]

5

Great. Thanks very much.

6

Your expressions are not equivalent over the complexes:

Sqrt[(a^2 + b^2)^7] == (a^2 + b^2)^(7/2) /. {a -> I, b -> I}
{Sqrt[(a^2 + b^2)^7], (a^2 + b^2)^(7/2)} /. {a -> I, b -> I}
7

I used assumptions that both a and b are non-negative real numbers. I still needed to use PowerExpand to make it work. Thanks for the feedback.

8

Using Simplify with assumptions :

Simplify[{Sqrt[(a^2+b^2)^7],(a^2+b^2)^(7/2)},Element[{a,b},NonNegativeReals]]
9

Ohh.. I like that. A better demonstration. Thanks Hans Milton.