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When can, or can't, terms be cancelled.

ie: $\frac{3x^2-1}{x^2}$

$x^2$ cannot be cancelled. Why not, and what are the rules?

dsDoan
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    What do you mean when you say "cancelled"? – Pedro Feb 27 '14 at 02:57
  • By "cancelled," I mean when can like terms, one on top, one on bottom, be divided into 1? – dsDoan Feb 27 '14 at 02:57
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    You're asking why $\frac{3x^2-1}{x^2}\neq \frac{3-1}{1}$? It's quite self evident, right? =) – Pedro Feb 27 '14 at 02:58
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    The point is $3x^2-1$ is not divisible by $x^2$. What you can do is write $$\frac{3x^2-1}{x^2}=3-\frac 1{x^2}$$ That's fine. – Pedro Feb 27 '14 at 03:00
  • Looking at your equation makes sense, but I'm trying to define a rule to myself. Instead of "cancel," I should have said "reduce."

    Something like: "The entire top term must be divisible by the entire bottom term (or vice versa) to reduce or 'cancel like terms.'"

     – dsDoan Feb 27 '14 at 03:28
  • Your idea is fine. – Pedro Feb 27 '14 at 03:28

2 Answers2

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Since “canceling” means “dividing the numerator and the denominator by the same (non-zero) number” one might cancel the given fraction by $7$, e.g., which yields to $\dfrac{\dfrac{3x^2}{7}-\dfrac{1}{7}}{\dfrac{x^2}{7}},$ which is perfectly right and perfectly senseless either.

Moral: canceling means not simplifying per se. If simplifying is the goal, first factorize numerator and denominator and look for common factors.

Michael Hoppe
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  1. Anything trig over/above something not trig.

1.a. unlike trig functions.

  1. Anytime there exists a higher power of a variable above or below some smaller power.
  2. Anytime there is a constant above or below with addition or subtraction operators with variables.

And of course this is all given that a series of variables with powers doesn't equal the below or above.

T.Woody
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