Question 176405
To simplify this expression, we need to simplify the individual radicals.


Let's simplify {{{sqrt(16x^3y^2)}}}



{{{sqrt(16x^3y^2)}}} Start with the given expression.



{{{sqrt(16x^2*x*y^2)}}} Factor {{{x^3}}} to get {{{x^2*x}}}



{{{sqrt(16)*sqrt(x^2)*sqrt(x)*sqrt(y^2)}}} Break up the square root.



{{{4*sqrt(x^2)*sqrt(x)*sqrt(y^2)}}} Take the square root of 16 to get 4



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



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



{{{4xy*sqrt(x)}}} Rearrange the terms.



So {{{sqrt(16x^3y^2)=4xy*sqrt(x)}}} where every variable is positive.



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Now let's simplify {{{sqrt(4x^3)}}}



{{{sqrt(4x^3)}}} Start with the given expression.



{{{sqrt(4x^2*x)}}} Factor {{{x^3}}} to get {{{x^2*x}}}



{{{sqrt(4)*sqrt(x^2)*sqrt(x)}}} Break up the square root.



{{{2*sqrt(x^2)*sqrt(x)}}} Take the square root of 4 to get 2



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



So {{{sqrt(4x^3)=2x*sqrt(x)}}} where "x" is positive.




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Since {{{sqrt(16x^3y^2)=4xy*sqrt(x)}}} and {{{sqrt(4x^3)=2x*sqrt(x)}}} (where every variable is positive), this means that 



{{{4*sqrt(16x^3y^2)-5y*sqrt(4x^3)}}}



simplifies to 



{{{4*(4xy*sqrt(x))-5y*(2x*sqrt(x))}}}



{{{16xy*sqrt(x)-5y*(2x*sqrt(x))}}} Multiply 4 and 4xy to get 16xy



{{{16xy*sqrt(x)-10xy*sqrt(x)}}} Multiply 5y and 2x to get 10xy



{{{sqrt(x)*(16xy-10xy)}}} Factor out the GCF {{{sqrt(x)}}}



{{{sqrt(x)*(6xy)}}} Combine like terms



{{{6xy*sqrt(x)}}} Rearrange the terms




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Answer:



So {{{4*sqrt(16x^3y^2)-5y*sqrt(4x^3)=6xy*sqrt(x)}}} where every variable is positive.