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Question: Can simplify be improved on this expression with square roots (or are the assumptions insufficient)?

Question:Can simplify be improved on this expression with square roots (or are the assumptions insufficient)?

C_R 3477 Maple
radical simplify assumptions simplification
5 hours ago
1

In the below I had to add the assumption x>=0 to get simplifications. Am I wrong with my interpretation that the other assumptions should have been sufficient?

(I would also be interested in any other ways to simplify this expression.)

NULL

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sqrt((-alpha^2*x^2 + R^2 + x^2)/(-alpha^2*x^2 + R^2))/sqrt(R^4/((-alpha^2*x^2 + R^2 + x^2)^2*(-alpha^2*x^2 + R^2)))

((-alpha^2*x^2+R^2+x^2)/(-alpha^2*x^2+R^2))^(1/2)/(R^4/((-alpha^2*x^2+R^2+x^2)^2*(-alpha^2*x^2+R^2)))^(1/2)

 (1)
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simplify((1),radical);

simplify/radical does not work because the signs of the nominators and denominators are not known

((-alpha^2*x^2+R^2+x^2)/(-alpha^2*x^2+R^2))^(1/2)/(R^4/((-alpha^2*x^2+R^2+x^2)^2*(-alpha^2*x^2+R^2)))^(1/2)

 (2)
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simplify((1),radical)assuming R>0,alpha>0,x*alpha<R

Since R and alpha are implicitly assumed real by R>0 and alpha>0, x must be real as well. However simplify does not deduce this form the assumptions:

((-alpha^2*x^2+R^2+x^2)/(-alpha^2*x^2+R^2))^(1/2)*(-alpha^2*x^2+R^2+x^2)/(R^2*signum(-alpha^2*x^2+R^2+x^2)*(1/(-alpha^2*x^2+R^2))^(1/2))

 (3)
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simplify((1),radical)assuming R>0,alpha>0,x*alpha<R,x::real

Assuming explicitly x::real has no effect

((-alpha^2*x^2+R^2+x^2)/(-alpha^2*x^2+R^2))^(1/2)*(-alpha^2*x^2+R^2+x^2)/(R^2*signum(-alpha^2*x^2+R^2+x^2)*(1/(-alpha^2*x^2+R^2))^(1/2))

 (4)
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simplify((1),radical)assuming R>0,alpha>0,x*alpha<R,x::nonnegative

Only when x>=0 is assumed the signum function disappears

(-alpha^2*x^2+R^2+x^2)*((-alpha^2*x^2+R^2+x^2)*(-alpha^2*x^2+R^2))^(1/2)/(R^2*(-alpha^2*x^2+R^2)^(1/2))

 (5)

Still Maple does not realise that -alpha^2*x^2+R^2is positive and can therefore be extracted from the root in the numerator to further simplify.
To do so annother simplify option is required (which still requires x>=0)

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simplify((1),power)assuming R>0,x>=0,alpha>0,x*alpha<R

(-alpha^2*x^2+R^2+x^2)^(3/2)/R^2

 (6)
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NULL


 

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