Question 200354
I'm assuming you meant to say "simplify"




{{{sqrt(25*a^12*b^8*c^12)}}} Start with the given expression.



{{{sqrt(25*a^2*a^2*a^2*a^2*a^2*a^2*b^8*c^12)}}} Factor {{{a^12}}} into {{{a^2*a^2*a^2*a^2*a^2*a^2}}}



{{{sqrt(25*a^2*a^2*a^2*a^2*a^2*a^2*b^2*b^2*b^2*b^2*c^12)}}} Factor {{{b^8}}} into {{{b^2*b^2*b^2*b^2}}}



{{{sqrt(25*a^2*a^2*a^2*a^2*a^2*a^2*b^2*b^2*b^2*b^2*c^2*c^2*c^2*c^2*c^2*c^2)}}} Factor {{{c^12}}} into {{{c^2*c^2*c^2*c^2*c^2*c^2}}}



{{{sqrt(25)*sqrt(a^2)*sqrt(a^2)*sqrt(a^2)*sqrt(a^2)*sqrt(a^2)*sqrt(a^2)*sqrt(b^2)*sqrt(b^2)*sqrt(b^2)*sqrt(b^2)*sqrt(c^2)*sqrt(c^2)*sqrt(c^2)*sqrt(c^2)*sqrt(c^2)*sqrt(c^2)}}} Break up the square root using the identity {{{sqrt(A*B)=sqrt(A)*sqrt(B)}}}.



{{{5*sqrt(a^2)*sqrt(a^2)*sqrt(a^2)*sqrt(a^2)*sqrt(a^2)*sqrt(a^2)*sqrt(b^2)*sqrt(b^2)*sqrt(b^2)*sqrt(b^2)*sqrt(c^2)*sqrt(c^2)*sqrt(c^2)*sqrt(c^2)*sqrt(c^2)*sqrt(c^2)}}} Take the square root of {{{25}}} to get {{{5}}}.



{{{5*a*a*a*a*a*a*sqrt(b^2)*sqrt(b^2)*sqrt(b^2)*sqrt(b^2)*sqrt(c^2)*sqrt(c^2)*sqrt(c^2)*sqrt(c^2)*sqrt(c^2)*sqrt(c^2)}}} Take the square root of {{{a^2}}} to get {{{a}}}.



{{{5*a*a*a*a*a*a*b*b*b*b*sqrt(c^2)*sqrt(c^2)*sqrt(c^2)*sqrt(c^2)*sqrt(c^2)*sqrt(c^2)}}} Take the square root of {{{b^2}}} to get {{{b}}}.



{{{5*a*a*a*a*a*a*b*b*b*b*c*c*c*c*c*c}}} Take the square root of {{{c^2}}} to get {{{c}}}.



{{{5a^6b^4c^6}}} Multiply


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Answer:



So {{{sqrt(25*a^12*b^8*c^12)}}} simplifies to {{{5a^6b^4c^6}}}



In other words, {{{sqrt(25*a^12*b^8*c^12)=5a^6b^4c^6}}} where every variable is non-negative.