Simplifying fractions with roots

[username12345]

Homework Statement

Simplify $$\frac{x^2 - \sqrt{x}}{\sqrt{x^5}}$$

Homework Equations

Unsure

The Attempt at a Solution

Tried to factorise the numerator and denominator. Not sure how to proceed given the subtraction in the numerator. Best effort so far:

$$<br /> \frac{x^2}{\sqrt{x^5}} - \frac{\sqrt{x}}{\sqrt{x^5} }} = <br /> \frac{x^2}{x^{ \frac{5}{2}}} - \frac{x^{\frac{1}{2}}} {x^\frac{5}{2}} =<br /> x^{ -\frac{1}{2}} - x^{-2} = <br /> \frac{1}{x^2} - \frac{1} {\sqrt{x}}$$

which, doesn't seem like much progress from the original equation

 

[Mark44]

If you multiplied top and bottom by x^(4/5), you'd at least get the radical out of the denominator, which is probably a good thing...

 

[username12345]

If you multiplied top and bottom by x^(4/5), you'd at least get the radical out of the denominator

why did you choose $$x^{\frac{4}{5}}$$, or more specifically, how did you decide that value?

 

[HallsofIvy]

yes, that was my question! I would use $x^{4/5}= \sqrt[5]{x^4}$ if I wanted to rationalize $\sqrt[5]{x}$, but this was $\sqrt{x^5}$. Why not multiply numerator and denominator by $\sqrt{x^5}$?

 

[username12345]

Using $$\sqrt{x^5}$$ in the numerator and denominator sets it up as $$\frac{ (x^2 - x^\frac{1}{2}) x^\frac{5}{2} } { x^\frac{5}{2} x^\frac{5}{2} }$$ and I end up with $$x^{-\frac{1}{2}} - x^{-2}$$. Am I starting off correctly?

 

[Mark44]

why did you choose $$x^{\frac{4}{5}}$$, or more specifically, how did you decide that value?

My mistake. I must have looked at the square root of x^5, and mentally translated it as x^(1/5). Sorry about that.