Solutions to Questions on Greatest Common Factor of Monomials

The detailed solutions and full explanations to the questions on Greatest Common Factor of Monomials are presented.

Questions with Solutions

1. Find the greatest common factor of the monomials 36 x² and 42 x³.
   Solution
   Write the prime factorization of the monomial 36 x²
   36 x² = 2 × 2 × 3 × 3 × x × x
   Write the prime factorization of the monomial 42 x³
   42 x³ = 2 × 3 × 7 × x × x × x
   The greatest common factor of 36 x² and 42 x³ is
   2 × 3 × x × x = 6 x ²


2. Find the greatest common factor of 45 x³, 60 x² and 75 x⁴.
   Solution
   Write the prime factorization of the monomial 45 x³
   45 x³ = 3 × 3 × 5 × x × x × x
   Write the prime factorization of the monomial 60 x²
   60 x² = 2 × 2 × 3 × 5 × x × x
   Write the prime factorization of the monomial 75 x⁴
   75 x⁴ = 3 × 5 × 5 × x × x × x × x
   The greatest common factor of 45 x³, 60 x² and 75 x⁴ is
   3 × 5 × x × x = 15 x ²


3. What is the greatest common factor of 50 x² y³ , 75 x² y² and 125 x⁴ y³?
   Solution
   Write the prime factorization of the monomial 50 x² y³
   50 x² y³ = 2 × 5 × 5 × x × x × y × y × y
   Write the prime factorization of the monomial 75 x² y²
   75 x² y² = 3 × 5 × 5 × x × x × y × y Write the prime factorization of the monomial 125 x⁴ y³
   125 x⁴ y³ = 5 × 5 × 5 × x × x × x × x × y × y × y
   The greatest common factor of 50 x² y³ , 75 x² y² and 125 x⁴ y³ is
   5 × 5 × x × x × y × y = 25 x ² y ²


4. 
   a) Find the prime factorization of the monomials of 35 x³ y ² and 42 x² y ³.
   b) Simplify the rational expression ( 35 x³ y ² ) / (42 x² y ³)
   Solution
   a) Write the prime factorization of the monomial 35 x³ y ²
   35 x³ y ² = 5 × 7 × x × x × x × y × y
   Write the prime factorization of the monomial 42 x² y ³
   42 x² y ³ = 2 × 3 × 7 × x × x × y × y × y
   We now use the prime factorization to simplify the rational expression
   ( 35 x³ y ² ) / (42 x² y ³) = (5 × 7 × x × x × x × y × y) / ( 2 × 3 × 7 × x × x × y × y × y) = 5x / 6y

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