## Intermediate Algebra (12th Edition)

$12km^2\left( m-2k^{2}+3km^{2}-5k^{3}m \right)$
$\bf{\text{Solution Outline:}}$ Get the $GCF$ of the given expression, $12km^3-24k^3m^2+36k^2m^4-60k^4m^3 .$ Divide the given expression and the $GCF.$ Express the answer as the product of the $GCF$ and the resulting quotient. $\bf{\text{Solution Details:}}$ The $GCF$ of the constants $( 12,-24,36,60 )$ is $12 .$ The $GCF$ of the common variable/s is the variable/s with the lowest exponent. Hence, the $GCF$ of the common variables $( km^3,k^3m^2,k^2m^4,k^4m^3 )$ is $km^2 .$ Hence, the entire expression has $GCF= 12km^2 .$ Factoring the $GCF= 12km^2 ,$ the given expression is equivalent to \begin{array}{l}\require{cancel} 12km^2\left( \dfrac{12km^3}{12km^2}-\dfrac{24k^3m^2}{12km^2}+\dfrac{36k^2m^4}{12km^2}-\dfrac{60k^4m^3}{12km^2} \right) .\end{array} Using the Quotient Rule of the laws of exponents which states that $\dfrac{x^m}{x^n}=x^{m-n},$ the expression above simplifies to \begin{array}{l}\require{cancel} 12km^2\left( k^{1-1}m^{3-2}-2k^{3-1}m^{2-2}+3k^{2-1}m^{4-2}-5k^{4-1}m^{3-2} \right) \\\\= 12km^2\left( k^{0}m^{1}-2k^{2}m^{0}+3k^{1}m^{2}-5k^{3}m^{1} \right) \\\\= 12km^2\left( (1)m-2k^{2}(1)+3km^{2}-5k^{3}m \right) \\\\= 12km^2\left( m-2k^{2}+3km^{2}-5k^{3}m \right) .\end{array}