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chwala

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- Homework Statement
- Evaluate ##\dfrac {x^\frac{3}{2}+xy}{xy-y^3}-\dfrac {\sqrt x}{\sqrt x -y}##

- Relevant Equations
- working with surds

My approach;

##\dfrac {x^\frac{3}{2}+xy}{xy-y^3}-\dfrac {\sqrt x}{\sqrt x -y}##

##\dfrac {x(\sqrt x+y)}{y(x-y^2)}-\dfrac {\sqrt x}{\sqrt x -y}=\dfrac{x(\sqrt x+y)(\sqrt x-y)-y\sqrt x(x-y^2)}{y(x-y^2)(\sqrt x-y)}=\dfrac{x(x-y\sqrt x+y\sqrt x-y^2)-y\sqrt x(x-y^2)}{y(x-y^2)(\sqrt x-y)}=\dfrac{x(x-y^2)-y\sqrt x(x-y^2)}{y(x-y^2)(\sqrt x-y)}=\dfrac{x-y\sqrt x}{y(\sqrt x-y)}##

Now on factorization i ended up with;

##\dfrac{x-y\sqrt x}{y(\sqrt x-y)}=\dfrac{\sqrt x(\sqrt x-y)}{y(\sqrt x -y)}=\dfrac {\sqrt x}{y}## which is correct as per the textbook solution. I would appreciate an alternative approach guys.

##\dfrac {x^\frac{3}{2}+xy}{xy-y^3}-\dfrac {\sqrt x}{\sqrt x -y}##

##\dfrac {x(\sqrt x+y)}{y(x-y^2)}-\dfrac {\sqrt x}{\sqrt x -y}=\dfrac{x(\sqrt x+y)(\sqrt x-y)-y\sqrt x(x-y^2)}{y(x-y^2)(\sqrt x-y)}=\dfrac{x(x-y\sqrt x+y\sqrt x-y^2)-y\sqrt x(x-y^2)}{y(x-y^2)(\sqrt x-y)}=\dfrac{x(x-y^2)-y\sqrt x(x-y^2)}{y(x-y^2)(\sqrt x-y)}=\dfrac{x-y\sqrt x}{y(\sqrt x-y)}##

Now on factorization i ended up with;

##\dfrac{x-y\sqrt x}{y(\sqrt x-y)}=\dfrac{\sqrt x(\sqrt x-y)}{y(\sqrt x -y)}=\dfrac {\sqrt x}{y}## which is correct as per the textbook solution. I would appreciate an alternative approach guys.

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