Simplify.

`sqrt((25x^2y)/(75xy^5))`

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`sqrt((25x^2y)/(75xy^5))=sqrt((25/75)(x^2/x)(y/y^5))=` `sqrt(1/3xy^(-4))` `=sqrt(1/3x/y^4)` `=sqrt(3)/(3y^2)sqrt(x)`

`sqrt((25x^2y)/(75xy^5))`

To simplify this expression, reduce the fraction 25/75 to its lowest term.

`=sqrt(1/3*(x^2y)/(xy^5))`

Then, to divide same variables, apply this property of exponents which is `a^m/a^n=a^(m-n)` .

`=sqrt(1/3x^(2-1)y^(1-5))=sqrt(1/3xy^(-4))`

For negative exponent, use `a^(-m)=1/a^m` .

`=sqrt(1/3x*1/y^4)=sqrt(x/(3y^4))`

Then, apply this property of radicals which is `root(n)(a/b)=root(n)(a)/root(n)(b)` .

`=sqrt(x)/sqrt(3y^4)`

Note that `root(m)(a^m)=a` . So,

`=sqrt(x)/sqrt(3y^2*y^2)= sqrt(x)/(y*ysqrt3)=sqrt(x)/(y^2sqrt3)`

To simplify further, rationalize the denominator.

`=sqrt(x)/(y^2sqrt3)*sqrt3/sqrt3`

Since `sqrta*sqrta=a` , then:

`=(sqrt3)(sqrtx)/(3y^2)`

In multiplying radicals with same index, apply this property `root(n)(a)*root(n)(b)=root(n)(a*b)` .

`=sqrt(3x)/(3y^2)`

**Hence, `sqrt((25x^2y)/(75xy^5))=sqrt(3x)/(3y^2)` .**

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