Question 176549
First, simplify {{{sqrt(3x^2y^3)}}}



{{{sqrt(3x^2y^3)=sqrt(3x^2y^2*y)=sqrt(3)*sqrt(x^2)*sqrt(y^2)*sqrt(y)=sqrt(3)*x*y*sqrt(y)=xy*sqrt(3y)}}}



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



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Now simplify {{{sqrt(5xy^3)}}}



{{{sqrt(5xy^3)=sqrt(5xy^2*y)=sqrt(5)*sqrt(x)*sqrt(y^2)*sqrt(y)=sqrt(5)*sqrt(x)*y*sqrt(y)=y*sqrt(5xy)}}}



So {{{sqrt(5xy^3)=y*sqrt(5xy)}}} where every variable is positive.



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So 


{{{sqrt(3x^2y^3)/(4*sqrt(5xy^3))}}} 



simplifies to



{{{(xy*sqrt(3y))/(4*y*sqrt(5xy))}}}



{{{(x*highlight(y)*sqrt(3y))/(4*highlight(y)*sqrt(5xy))}}} Highlight the common terms.



{{{(x*cross(y)*sqrt(3y))/(4*cross(y)*sqrt(5xy))}}} Cancel out the common terms.



{{{(x*sqrt(3y))/(4*sqrt(5xy))}}} Simplify



{{{(x*sqrt(3y)*sqrt(5xy))/(4*sqrt(5xy)*sqrt(5xy))}}} Multiply the numerator and denominator by {{{sqrt(5xy)}}} (to rationalize the denominator)



{{{(x*sqrt(3y)*sqrt(5xy))/(4*5xy)}}} Multiply {{{sqrt(5xy)}}} by itself to get {{{sqrt(5xy)*sqrt(5xy)=(sqrt(5xy))^2=5xy}}}



{{{(x*sqrt(3y*5xy))/(4*5xy)}}} Combine the radicals



{{{(x*sqrt(15xy^2))/(20xy)}}} Multiply



{{{(cross(x)*sqrt(15xy^2))/(20*cross(x)y)}}} Cancel out like terms.



{{{sqrt(15xy^2)/(20y)}}} Simplify



{{{(y*sqrt(15x))/(20y)}}} Simplify the square root.



{{{(cross(y)*sqrt(15x))/(20cross(y))}}} Cancel out like terms.



{{{sqrt(15x)/20}}}




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Answer:



So after all of that, the solution is



{{{sqrt(3x^2y^3)/(4*sqrt(5xy^3))=sqrt(15x)/20}}} where every variable is positive