Simplify. `sqrt((25x^2y)/(75xy^5))` 

Expert Answers
lemjay eNotes educator| Certified Educator


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


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


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


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


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. 


Since `sqrta*sqrta=a` , then:


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


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

oldnick | Student

`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)`

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