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Finding The GCF and LCM of Terms with Variables

Date: 08/25/99 at 15:38:26
From: Amiri & Iverson
Subject: GCF and LCM of Terms with Variables

Thank you very much. I really appreciate all of the help you have 
given me.

Here are other questions I've had; maybe you could help me out.

1. (x^6)(xy^3)
   I understand the concept of exponents, but I don't understand what 
   I must do in order to simplify this problem.

2. (2rs^5)(-6mr^6)
   Same as above.

3. Find the greatest common factor: 12a^3c and 15ab^3
   How do you find the GCF of a number and two variables that have 

4. Find the least common multiple: 7x and 8x^2
   Last year in Pre-algebra we never reviewed least common multiples,
   and I have forgotten how to find them.

Thank you very much for the help.
-Shahrum Amiri

Date: 08/26/99 at 12:09:22
From: Doctor Peterson
Subject: Re: GCF and LCM of Terms with Variables

Hi, Shahrum.

1. (x^6)(xy^3)

First use the associative and commutative properties to get the x's 
and the y's together by dropping parentheses and moving powers if 

     (x^6)(x y^3) = (x^6*x)(y^3)

Now use the distributive property for exponents,

     n^a * n^b = n^(a+b)

remembering that x = x^1:

     x^6*x = x^6 * x^1 = x^(6+1) = x^7

Now put it together:

     x^7 y^3

If your problem is in the use of the distributive property itself, it 
may help just to picture it in terms of repeated multiplication:

     x^6 * x * y^3 = x*x*x*x*x*x * x * y*y*y
                   = x*x*x*x*x*x*x * y*y*y
                   = x^7 * y^3

It's really nothing more than counting factors.

2. (2rs^5)(-6mr^6)

The only difference here is that you have to reorder the parts, and 
that there are numeric coefficients involved. First pull it apart by 
dropping the parentheses

     2 r s^5 (-6) m r^6

then reorder (I like to put variables in alphabetical order to make 
sure I got them all)

     (2) (-6) m r r^6 s^5
     ======== = ===== ===

and finally combine numbers with numbers and powers of the same 
variable with one another as in the previous example

     -12 m r^7 s^5

and you're done.

3. Find the greatest common factor: 12a^3c and 15ab^3

As with numbers, one good way to find the GCF is to list the factors 
of the two expressions in the same order (I like numerical order for 
primes and alphabetical order for variables), and then choose the 
largest power of each variable that will divide both - that is, the 
smallest power present in each column of my diagram:

     12a^3c = 2^2 3     a^3     c
     15ab^3 =     3 5   a^1 b^3
              --- - -   --- --- -
                  3     a^1

The answer is 3a. The 3 is the GCF of 12 and 15, and the a is the only 
variable that is present in both expressions. The 2^2, 5, b^3, and c 
disappear because they each appear in only one of the expressions.

4. Find the least common multiple: 7x and 8x^2

Finding an LCM is the opposite of the GCF: list the factors, as above, 
but look for the _largest_ power in each column:

     7x   = 7     x
     8x^2 =   2^3 x^2
            - --- ---
            7 2^3 x^2 = 56x^2

This is because the LCM has to contain at least the 7 and one x, and 
also at least the 8 and two x's; that means it needs both the 7 and 
the 8, and two x's will include the one x.

Have you noticed, by the way, that for any numbers or expressions A 
and B,

     A * B = gcd(A,B) * lcm(A,B)

In this case, A = 7x, B = 8x^2, GCD = x, and LCM = 56x^2; the products 
on both sides are 56x^3. This lets you find the LCM if you know the 
GCD. It works because the LCM includes all the factors of A and B, and 
the GCD consists of the factors that have to be included twice to form 
the complete product. If you think through this concept, it may help 
you understand better what's going on when you find the GCD or LCM. 
This picture illustrates it; see if you can follow what I mean:

      A = a1 a2 a3 a4 x1 x2 x3
      B =             x1 x2 x3 b4 b5

    lcm = a1 a2 a3 a4 x1 x2 x3 b4 b5
    gcd =             x1 x2 x3

- Doctor Peterson, The Math Forum   
Associated Topics:
Middle School Algebra
Middle School Exponents

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