Question 178102

{{{sqrt(121*x^12*y^16*z^6)}}} Start with the given expression.



{{{sqrt(121*x^2*x^2*x^2*x^2*x^2*x^2*y^16*z^6)}}} Factor {{{x^12}}} into {{{x^2*x^2*x^2*x^2*x^2*x^2}}} (note: there are 6 {{{x^2}}} terms)



{{{sqrt(121*x^2*x^2*x^2*x^2*x^2*x^2*y^2*y^2*y^2*y^2*y^2*y^2*y^2*y^2*z^6)}}} Factor {{{y^16}}} into {{{y^2*y^2*y^2*y^2*y^2*y^2*y^2*y^2}}} (note: there are 8 {{{y^2}}} terms)



{{{sqrt(121*x^2*x^2*x^2*x^2*x^2*x^2*y^2*y^2*y^2*y^2*y^2*y^2*y^2*y^2*z^2*z^2*z^2)}}} Factor {{{z^6}}} into {{{z^2*z^2*z^2}}} (note: there are 3 {{{z^2}}} terms)



{{{sqrt(121)*sqrt(x^2)*sqrt(x^2)*sqrt(x^2)*sqrt(x^2)*sqrt(x^2)*sqrt(x^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(z^2)*sqrt(z^2)*sqrt(z^2)}}} Break up the square root using the identity {{{sqrt(A*B)=sqrt(A)*sqrt(B)}}}.



{{{11*sqrt(x^2)*sqrt(x^2)*sqrt(x^2)*sqrt(x^2)*sqrt(x^2)*sqrt(x^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(z^2)*sqrt(z^2)*sqrt(z^2)}}} Take the square root of {{{121}}} to get {{{11}}}.



{{{11*x*x*x*x*x*x*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(y^2)*sqrt(z^2)*sqrt(z^2)*sqrt(z^2)}}} Take the square root of {{{x^2}}} to get {{{x}}}.



{{{11*x*x*x*x*x*x*y*y*y*y*y*y*y*y*sqrt(z^2)*sqrt(z^2)*sqrt(z^2)}}} Take the square root of {{{y^2}}} to get {{{y}}}.



{{{11*x*x*x*x*x*x*y*y*y*y*y*y*y*y*z*z*z}}} Take the square root of {{{z^2}}} to get {{{z}}}.



{{{11x^6y^8z^3}}} Multiply and simplify.


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Answer:



So {{{sqrt(121*x^12*y^16*z^6)}}} simplifies to {{{11x^6y^8z^3}}}



In other words, {{{sqrt(121*x^12*y^16*z^6)=11x^6y^8z^3}}} where every variable is positive.