1. **Stating the problem:** We want to find the greatest common monomial factor (GCMF) of given monomials. The GCMF is the largest monomial that divides each of the given monomials exactly. 2. **Formula and rules:** - A monomial is a product of a number (coefficient) and variables with non-negative integer exponents, e.g., $6x^3y^2$. - To find the GCMF of monomials, find the greatest common divisor (GCD) of the coefficients and the lowest powers of each variable common to all monomials. 3. **Example 1:** Find the GCMF of $12x^3y^2$ and $18x^2y^4$. - Coefficients: GCD of 12 and 18 is $6$. - Variables: For $x$, powers are 3 and 2, so take the minimum $2$. - For $y$, powers are 2 and 4, so take the minimum $2$. - Therefore, GCMF is $6x^2y^2$. 4. **Example 2:** Find the GCMF of $8a^4b^3$ and $12a^2b^5$. - Coefficients: GCD of 8 and 12 is $4$. - Variables: For $a$, powers are 4 and 2, minimum is $2$. - For $b$, powers are 3 and 5, minimum is $3$. - So, GCMF is $4a^2b^3$. 5. **Summary:** To find the greatest common monomial factor, find the GCD of coefficients and take the smallest exponent for each variable present in all monomials. This method helps simplify expressions and factor polynomials efficiently.