Question 177297
First, simplify {{{sqrt(50zu^2)}}}


{{{sqrt(50*z*u^2)}}} Start with the given expression.



{{{sqrt(25*2*z*u^2)}}} Factor {{{50}}} into {{{25*2}}}



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



{{{5*sqrt(2)*sqrt(z)*sqrt(u^2)}}} Take the square root of {{{25}}} to get {{{5}}}.



{{{5*sqrt(2)*sqrt(z)*u}}} Take the square root of {{{u^2}}} to get {{{u}}}.



{{{5u*sqrt(2z)}}} Rearrange and combine the terms.




So {{{sqrt(50*z*u^2)=5u*sqrt(2z)}}} where every variable is positive.



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Now let's simplify {{{sqrt(2*z^3)}}}



{{{sqrt(2*z^3)}}} Start with the given expression.



{{{sqrt(2*z^2*z)}}} Factor {{{z^3}}} into {{{z^2*z}}}



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



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



{{{z*sqrt(2z)}}} Rearrange and combine the terms.



So {{{sqrt(2*z^3)=z*sqrt(2z)}}} where every variable is positive.




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So 



{{{z*sqrt(50zu^2)-8u*sqrt(2*z^3)}}} 



simplifies to 



{{{z*5u*sqrt(2z)-8u*z*sqrt(2z)}}}




{{{5uz*sqrt(2z)-8uz*sqrt(2z)}}} Rearrange the terms.



{{{sqrt(2z)*(5uz-8uz)}}} Factor out {{{sqrt(2z)}}}



{{{sqrt(2z)*(-3uz)}}} Combine like terms.



{{{-3uz*sqrt(2z)}}} Rearrange the terms.




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Answer:


So {{{z*sqrt(50zu^2)-8u*sqrt(2*z^3)}}} completely simplifies to {{{-3uz*sqrt(2z)}}}




In other words, {{{z*sqrt(50zu^2)-8u*sqrt(2*z^3)=-3uz*sqrt(2z)}}} where every variable is positive.