Simplifying a rational expression

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Discussion Overview

The discussion revolves around the simplification of a rational expression involving square roots and cancellation of terms. Participants explore the implications of simplifying the expression step by step, particularly focusing on the treatment of the square root of a squared term.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant asks whether the term ##\sqrt{(x-2)^{2}}## can be canceled with the ##(x-2)## in the denominator.
  • Some participants assert that cancellation is permissible.
  • Others caution that the square root of a squared term is not always equal to the original term due to sign considerations, emphasizing the importance of evaluating specific values like ##x=0## to illustrate potential discrepancies.
  • A participant mentions that ##\sqrt{x^2} = |x|##, reinforcing the need to consider absolute values in simplifications.
  • Another participant points out that the expression in question does not qualify as a "rational function."

Areas of Agreement / Disagreement

Participants express disagreement regarding the cancellation of terms involving square roots, with some supporting cancellation and others highlighting the need for caution due to sign issues. The discussion remains unresolved with competing views on the matter.

Contextual Notes

Participants note that the treatment of the square root introduces conditions based on the values of ##x##, which may affect the validity of simplifications. The discussion also touches on the definition of rational functions, indicating a potential misunderstanding of terminology.

Mr Davis 97
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Given that we have the expression ##\displaystyle-\frac{1}{(x-2)(x-2)(x-3)}~\cdot~\sqrt{\frac{(x-2)^{2}}{(x-3)(x-1)}} ##, how do we simplify it, step by step? Specifically, I am concerned about the ##\sqrt{(x-2)^{2}}## term. Are we allowed to cancel this with the ##(x-2)## in the denominator?
 
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Yes.
 
paisiello2 said:
Yes.
I disagree. You have to be careful about signs. The square root of x squared is not always equal to x.

Take, for instance, x=0 and evaluate the given expression before and after cancellation. Do the two give the same result?
 
jbriggs444 said:
I disagree. You have to be careful about signs. The square root of x squared is not always equal to x.

Take, for instance, x=0 and evaluate the given expression before and after cancellation. Do the two give the same result?
I agree with jbriggs444 here, and would add that ##\sqrt{x^2} = |x|##, which is something I mentioned in your other thread on rational expressions.
 
By the way- this is not a "rational function"!
 

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