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I am continually running into the problem where square roots defeat any attempt to simplify an expression. For example:

sqrt(-x_3 + sqrt(x_1^2 + x_2^2 + x_3^2))*sqrt(x_3 + sqrt(x_1^2 + x_2^2 + x_3^2))/sqrt(x_2^2/x_1^2 + 1)

should simplify to x_1.

I understand the reason sage hesitates to simplify a square root. But being unable to force the issue is starting to render sage almost completely useless.

Does anybody know how to get sage to simplify these types of expressions? simplify_radical does not work. The closest I can come is to square the expression first, then simplify and take the square root. That still does not get rid of the final square root. I wonder if making the expression complex might work, and if so, how might I do that?

I am continually running into the problem where square roots defeat any attempt to simplify an expression. For example:

sqrt(-x_3 + sqrt(x_1^2 + x_2^2 + x_3^2))*sqrt(x_3 + sqrt(x_1^2 + x_2^2 +
x_3^2))/sqrt(x_2^2/x_1^2 + 1)
1)