What is the greatest common factor of 36x^3y^2 and 48x^2y?

To find the greatest common factor (GCF) of the expressions ( 36x^3y^2 ) and ( 48x^2y ), it's essential to consider both the numerical coefficients and the variables separately.

First, let's break down the coefficients.

• The number 36 can be factored into primes as ( 36 = 2^2 \times 3^2 ).

• The number 48 can be factored into primes as ( 48 = 2^4 \times 3^1 ).

Next, to find the GCF of the coefficients, we take the minimum exponent for each prime factor:

• For the factor of 2, the minimum between ( 2^2 ) (from 36) and ( 2^4 ) (from 48) is ( 2^2 ).

• For the factor of 3, the minimum between ( 3^2 ) (from 36) and ( 3^1 ) (from 48) is ( 3^1 ).

Combining these gives us the GCF of the numbers:

( 2^2 \times 3^1 = 4 \times