Identify the greatest common factor of \( 50 a^{4} c^{3} z^{3} \) and \( 25 c^{2} w^{3} z^{2} \)

To identify the greatest common factor (GCF) of the two expressions \( 50 a^{4} c^{3} z^{3} \) and \( 25 c^{2} w^{3} z^{2} \), we first break down each expression into its prime factor components. For \( 50 a^{4} c^{3} z^{3} \): - The numeric coefficient is \(50\), which can be factored into \(2 \times 5^2\). - The variable \(a^{4}\) has nothing in common with the second expression. - The variable \(c^{3}\) means we take \(c^{3}\). - The variable \(z^{3}\) means we take \(z^{3}\). For \( 25 c^{2} w^{3} z^{2} \): - The numeric coefficient is \(25\), which factors into \(5^2\). - The variable \(c^{2}\) means we take \(c^{2}\). - The variable \(z^{2}\) means we take \(z^{2}\). Now we find the GCF by taking the minimum power of each common factor: - For the numeric coefficients: \( \min(2 \times 5^2, 5^2) = 5^2 = 25\). - For \(c\): \( \min(c^{3}, c^{2}) = c^{2}\). - For \(z\): \( \min(z^{3}, z^{2}) = z^{2}\). - Variable \(a\) and \(w\) do not contribute as they are not common. Thus, the greatest common factor is: \[ 25 c^{2} z^{2} \]