How do I find the GCF of algebraic expressions involving negative exponents? I'm currently reviewing college algebra and I'm learning about factoring polynomials and algebraic expressions.
I have no difficulties finding the GCF of algebraic expressions whose variables have positive integer exponents, but I don't understand the process when it comes to algebraic expressions whose variables have negative exponents.
I understand why you factor out the power of each of the variables with the smallest exponent when working with positive exponents, but I don't see why that rule applies when dealing with fractional and negative exponents. Actually, I don't have difficulty seeing why the rule applies to positive fractional exponents because, for instance, I can see that $$3x^{3/2}-9x^{1/2}+6x^{1/2}=3(x^{1/2})^3-9(x^{1/2})^1+6(x^{1/2})^1$$ and so I can see that the GCF is $3(x^{1/2})^1=3x^{1/2}$. But admittedly, I did not see this expression and know intuitively that the GCF should be the expression that contained the power of $x$ with the smallest exponent. It wasn't until I rewrote the expression as above that I saw that why the rule makes sense.

How can I rewrite expressions involving variables with negative exponents to see why the rule is still valid? If the terms in the above example were instead
$$3x^{2/7}-9x^{-3/4}+6x^{-3/5}$$ would the GCF be $3x^{-3/4}$ as the rule would suggest?

What is the definition of greatest common factor I should keep in mind when dealing with algebraic expressions and polynomials?

It hinges on the question of whether you consider $x^{-3/5}$ to be a factor of $x^{2/7}$, which I suspect varies from source to source. The advantage of being able to factor out a term with negative exponent like this is that it leaves a polynomial with positive exponents, which you're more likely to be able to do something with.
