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Simplifying fractions with roots

1. Homework Statement

Simplify [tex]\frac{x^2 - \sqrt{x}}{\sqrt{x^5}} [/tex]



2. Homework Equations

Unsure

3. 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:

[tex]

\frac{x^2}{\sqrt{x^5}} - \frac{\sqrt{x}}{\sqrt{x^5} }} =
\frac{x^2}{x^{ \frac{5}{2}}} - \frac{x^{\frac{1}{2}}} {x^\frac{5}{2}} =
x^{ -\frac{1}{2}} - x^{-2} =
\frac{1}{x^2} - \frac{1} {\sqrt{x}}
[/tex]

which, doesn't seem like much progress from the original equation
 
32,575
4,305
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...
 
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 [tex]x^{\frac{4}{5}}[/tex], or more specifically, how did you decide that value?
 

HallsofIvy

Science Advisor
Homework Helper
41,682
864
yes, that was my question! I would use [itex]x^{4/5}= \sqrt[5]{x^4}[/itex] if I wanted to rationalize [itex]\sqrt[5]{x}[/itex], but this was [itex]\sqrt{x^5}[/itex]. Why not multiply numerator and denominator by [itex]\sqrt{x^5}[/itex]?
 
Using [tex]\sqrt{x^5}[/tex] in the numerator and denominator sets it up as [tex]\frac{ (x^2 - x^\frac{1}{2}) x^\frac{5}{2} } { x^\frac{5}{2} x^\frac{5}{2} } [/tex] and I end up with [tex]x^{-\frac{1}{2}} - x^{-2} [/tex]. Am I starting off correctly?
 
32,575
4,305
why did you choose [tex]x^{\frac{4}{5}}[/tex], 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.
 

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