Simplifying fractions with roots

In summary, the conversation is about simplifying the expression \frac{x^2 - \sqrt{x}}{\sqrt{x^5}}, with the attempt at factoring and multiplying by x^(4/5) to rationalize the denominator. However, there is some confusion over the choice of x^(4/5) and how to proceed with the simplification.
  • #1
username12345
48
0

Homework Statement



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



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:

[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
 
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  • #2
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...
 
  • #3
Mark44 said:
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?
 
  • #4
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]?
 
  • #5
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?
 
  • #6
username12345 said:
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.
 

What is a fraction with a root?

A fraction with a root is a fraction where either the numerator or denominator (or both) contains a square root or other type of root.

Why do we simplify fractions with roots?

Simplifying fractions with roots makes them easier to work with and understand. It also helps to reduce the complexity of calculations involving these fractions.

How do we simplify fractions with roots?

To simplify a fraction with a root, we need to factor the number under the root and then cancel out any common factors between the numerator and denominator. This will result in the fraction being expressed in its simplest form.

Can we simplify fractions with different types of roots?

Yes, we can simplify fractions with any type of root, including square roots, cube roots, and higher order roots. The process is the same for all types of roots.

Are there any special rules for simplifying fractions with roots?

Yes, there are a few special rules to keep in mind when simplifying fractions with roots. For example, we cannot have a root in the denominator of a fraction, so we need to rationalize the denominator by multiplying by a conjugate. Additionally, if the root in the numerator is a perfect square, we can simplify it further by taking the root outside of the fraction.

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