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 \( 36x^2 \) and \( 42x^3 \).
    Solution
    Write the prime factorization of the monomial \( 36x^2 \):
    \[ 36x^2 = {2 \times 2 \times 3 \times 3} \times x \times x \] Write the prime factorization of the monomial \( 42x^3 \):
    \[ 42x^3 = {2 \times 3 \times 7} \times x \times x \times x \] The greatest common factor of \( 36x^2 \) and \( 42x^3 \) is:
    \[ {2 \times 3 \times x \times x = 6x^2} \]

  2. Find the greatest common factor of \( 45x^3 \), \( 60x^2 \), and \( 75x^4 \).

    Solution

    Write the prime factorization of the monomial \( 45x^3 \):

    \[ 45x^3 = \times 3 \times {5} \times {x \times x} \times x \]

    Write the prime factorization of the monomial \( 60x^2 \):

    \[ 60x^2 = 2 \times 2 \times \times {5} \times {x \times x} \]

    Write the prime factorization of the monomial \( 75x^4 \):

    \[ 75x^4 = {3} \times {5} \times 5 \times {x \times x} \times x \times x \]

    The greatest common factor of \( 45x^3 \), \( 60x^2 \), and \( 75x^4 \) is:

    \[ {3 \times 5 \times x \times x = 15x^2} \]

  3. What is the greatest common factor of \(50x^2y^3\), \(75x^2y^2\), and \(125x^4y^3\)?

    Solution

    Write the prime factorization of the monomial \(50x^2y^3\):

    \[ 50x^2y^3 = 2 \times 5 \times 5 \times x \times x \times y \times y \times y \]

    Write the prime factorization of the monomial \(75x^2y^2\):

    \[ 75x^2y^2 = 3 \times 5 \times 5 \times x \times x \times y \times y \]

    Write the prime factorization of the monomial \(125x^4y^3\):

    \[ 125x^4y^3 = 5 \times 5 \times 5 \times x \times x \times x \times x \times y \times y \times y \]

    The greatest common factor of \(50x^2y^3\), \(75x^2y^2\), and \(125x^4y^3\) is:

    \[ 5 \times 5 \times x \times x \times y \times y = 25x^2y^2 \]

More Questions with Solutions

  1. Find the prime factorization of the monomials of \( 35x^3y^2 \) and \( 42x^2y^3 \).
  2. Simplify the rational expression \( \dfrac{35x^3y^2}{42x^2y^3} \).

Solution

a) Write the prime factorization of the monomial \( 35x^3y^2 \):

\[ 35x^3y^2 = 5 \times 7 \times x \times x \times x \times y \times y \]

Now write the prime factorization of the monomial \( 42x^2y^3 \):

\[ 42x^2y^3 = 2 \times 3 \times 7 \times x \times x \times y \times y \times y \]

b) Use the prime factorizations to simplify the rational expression:

\[ \dfrac{35x^3y^2}{42x^2y^3} = \dfrac{5 {7} \times {x \times x} \times x \times {y \times y}}{2 \times 3 \times {7} \times {x \times x} \times{y \times y} \times y} \]

Cancel the common factors:

\[ = \dfrac{5x}{6y} \]

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