how to find LCM of a^2-a^2r^2 and a^2r^2

Question

I want to know how to find LCM of expressions without any numbers.

Answer 1

Note that the expression:

    a² - a²r²  is the difference of two squares and factors into:

    (a+ar)(a-ar)

    aa(1+r)(1-r)

 

The factors of:

   a²r²

 are:

   aarr

 

Just like finding the LCM after you have factored two numbers (that is, the LCM is composed of the least set of factors that include the factors of both numbers), the LCM is:

   aarr(1+r)(1-r)

    a²r²(1+r)(1-r)

 

To verify:   a²(1+r)(1-r) times r² equals a²r²(1+r)(1-r)

                  a²r²  times (1+r)(1-r) equals a²r²(1+r)(1-r)

 

Answer 2

Factor out the longer expression if possible.

 

a² - a²r² = a²(1 - r²) = a(1 + r)(1 - r)

 

 

Then the other factor is   a²r²

 

 

The LCM is the product of the factors.

 

 

LCM = a³(1 - r)(1 + r)r²