How do you simplify #sqrt(x^6*y^6)#?

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Answer 1 · 2015-07-23T02:14:44

First rewrite, as #x^6=(x^3)^2and y^6=(y^3)^2#

#=sqrt((x^3)^2*(y^3)^2)=sqrt((x^3)^2)*sqrt((y^3)^2))#

#=x^3*y^3=x^3y^3#

Answer 2 · 2015-07-31T06:14:27

#sqrt(x^6*y^6) = abs(x^3y^3)#

#sqrt(x^6*y^6) = sqrt((xy)^6) = sqrt(((xy)^3)^2) = abs((xy)^3) = abs(x^3y^3)#

using #sqrt(a^2) = abs(a)# for all #a in RR#.

Note especially that we need the absolute value to cover the case where #a < 0#, since #sqrt# denotes the positive square root.