Finding the HCF of Numerator and Denominator

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To determine if a numerator and denominator have a highest common factor (HCF), understanding common divisibility rules and complete factorization is essential. If no common factors exist, the fraction is already simplified. For more complex expressions involving multiple variables, methods such as grouping, inspection, and polynomial division may be necessary. Resources like basic mathematics textbooks or algebra courses can provide guidance on divisibility rules and factorization techniques. Mastering these concepts can streamline the process of simplifying fractions effectively.
Miike012
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After finding the simple factors in the numerator and denominator... Is there any way of knowing if the numerator and denominator has a HCF?

I think this would be very beneficial for me to know because if I want to simplify a "hard problem" into s simpler equation I can. However when I try to find HCF in some problems I find my self waisting time because most of the problems don't have a HCF, therefore can not be simplified.

So is there a way to quickley decide if there is an hcf? Are there any methods to doing this?

thank you.
Mike
 
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This is mostly a matter of knowing common divisibility rules and knowing how to, if possible, completely factor numbers. One of the goals is to know or find out if a fraction is already simplified --- as you found, if no common factors occur in both numerator and denominator, the fraction is simplified (assuming your interest is not about "mixed number" versus improper fraction).
 
Im not familliar with all of the divisibility rules... where can I find them?
 
Miike012 said:
Im not familliar with all of the divisibility rules... where can I find them?

I assumed your point of view was Basic Mathematics as in elementary basic education. Any beginning Arithmetic or other Basic Mathematics textbook would include divisibility rules. Is your point of view, instead something more advanced?
 
Im talking about a ratio that consists of more than one variable..
example..
(xy^5 - x^3 + z -10) / ( xyz 10x -9y ^2)...
I just made this up... but this is what I am talking about...
 
You want to factor rational expressions, so now we see. Use grouping, inspection, looking for any common factors among the numerator terms or the denominator terms; depending on the expressions present maybe try Rational Roots Theorem, if expressions are of the right form maybe try polynomial division, or if practical synthetic division. Some parts of Intermediate and College Algebra are dedicated to what you asked.
 

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