

Find the greatest common factor of the monomials 36 x^{2} and 42 x^{3}.
Solution
Write the prime factorization of the monomial 36 x^{2}
36 x^{2} = 2 × 2 × 3 × 3 × x × x
Write the prime factorization of the monomial 42 x^{3}
42 x^{3} = 2 × 3 × 7 × x × x × x
The greatest common factor of 36 x^{2} and 42 x^{3} is
2 × 3 × x × x = 6 x^{ 2}

Find the greatest common factor of 45 x^{3}, 60 x^{2} and 75 x^{4}.
Solution
Write the prime factorization of the monomial 45 x^{3}
45 x^{3} = 3 × 3 × 5 × x × x × x
Write the prime factorization of the monomial 60 x^{2}
60 x^{2} = 2 × 2 × 3 × 5 × x × x
Write the prime factorization of the monomial 75 x^{4}
75 x^{4} = 3 × 5 × 5 × x × x × x × x
The greatest common factor of 45 x^{3}, 60 x^{2} and 75 x^{4} is
3 × 5 × x × x = 15 x^{ 2}

What is the greatest common factor of 50 x^{2} y^{3} , 75 x^{2} y^{2} and 125 x^{4} y^{3}?
Solution
Write the prime factorization of the monomial 50 x^{2} y^{3}
50 x^{2} y^{3} = 2 × 5 × 5 × x × x × y × y × y
Write the prime factorization of the monomial 75 x^{2} y^{2}
75 x^{2} y^{2} = 3 × 5 × 5 × x × x × y × y
Write the prime factorization of the monomial 125 x^{4} y^{3}
125 x^{4} y^{3} = 5 × 5 × 5 × x × x × x × x × y × y × y
The greatest common factor of 50 x^{2} y^{3} , 75 x^{2} y^{2} and 125 x^{4} y^{3} is
5 × 5 × x × x × y × y = 25 x^{ 2} y^{ 2}

a) Find the prime factorization of the monomials of 35 x^{3} y^{ 2} and 42 x^{2} y^{ 3}.
b) Simplify the rational expression ( 35 x^{3} y^{ 2} ) / (42 x^{2} y^{ 3})
Solution
a) Write the prime factorization of the monomial 35 x^{3} y^{ 2}
35 x^{3} y^{ 2} = 5 × 7 × x × x × x × y × y
Write the prime factorization of the monomial 42 x^{2} y^{ 3}
42 x^{2} y^{ 3} = 2 × 3 × 7 × x × x × y × y × y
We now use the prime factorization to simplify the rational expression
( 35 x^{3} y^{ 2} ) / (42 x^{2} y^{ 3}) = (5 × 7 × x × x × x × y × y) / ( 2 × 3 × 7 × x × x × y × y × y)
= 5x / 6y

