Express each radical in simplest form, rationalize denominators and perform the indicated operations. `5sqrt(x^3y)-sqrt(4xy^5)`  Express each radical in simplest form, rationalize denominators...

Express each radical in simplest form, rationalize denominators and perform the indicated operations. 

`5sqrt(x^3y)-sqrt(4xy^5)` 

Express each radical in simplest form, rationalize denominators and perform the indicated operations. 

`sqrt(x/y^5)- sqrt(y/x^3)`

Express each radical in simplest form, rationalize denominators and perform the indicated operations. 

`root(3)(xy^-1)` - `root(3)(8x^(-2)y^2)`

Perform the indicated operations, expressing answers in simplest form with rationalized denominators. 

`sqrt(x)(sqrt(xy)+sqrt(z^3))`

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Asked on by lameduck27

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justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The following expressions have to be simplified and the denominator rationalized:

  •  `5*sqrt(x^3*y) - sqrt(4*x*y^5)`

= `5*sqrt(x*x^2*y) - sqrt(2^2*x*y^4*y)`

= `5*x*sqrt(x*y) - 2*y^2*sqrt(x*y)`

= `sqrt(xy)(5x - 2y^2)`

  • `sqrt(x/y^5) - sqrt(y/x^3)`

= `sqrt(x/(y*y^4)) - sqrt(y/(x*x^2))`

= `(1/y^2)*sqrt(x/y) - (1/x)*sqrt(y/x)`

= `(1/y^2)*sqrt(x/y)*(sqrt y/sqrt y) - (1/x)*sqrt(y/x)*(sqrtx/sqrt x)`

= `(1/y^2)*sqrt(x*y)/y - (1/x)*sqrt(x*y)/x`

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

  • `root(3)(xy^-1)-root(3)(8x^-2y^2)`

= `root(3)(x/y)-2*root(3)(y^2/x^2)`

= `(x^(1/3)/y^(1/3))-2*(y^(2/3)/x^(2/3))`

= `(x^(1/3)*y^(2/3))/y - 2*(y^(2/3)*x^(1/3))/x`

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

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

  • `sqrtx*(sqrt(xy) - sqrt(z^3))`

= `sqrtx*sqrt x*sqrt y - z*sqrt z*sqrt x`

= `x*sqrt y - z*sqrt*(x*z)`

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