Question 169942

{{{((z^2+8z+16)/(z^2+13z+36))/((z^2+4z)/(z^2+15z+54))}}} Start with the given expression.



{{{((z^2+8z+16)/(z^2+13z+36))((z^2+15z+54)/(z^2+4z))}}} Multiply the first fraction {{{(z^2+8z+16)/(z^2+13z+36)}}} by the reciprocal of the second fraction {{{(z^2+4z)/(z^2+15z+54)}}}.



{{{(((z+4)(z+4))/(z^2+13z+36))((z^2+15z+54)/(z^2+4z))}}} Factor {{{z^2+8z+16}}} to get {{{(z+4)(z+4)}}}.



{{{(((z+4)(z+4))/((z+9)(z+4)))((z^2+15z+54)/(z^2+4z))}}} Factor {{{z^2+13z+36}}} to get {{{(z+9)(z+4)}}}.

 

{{{(((z+4)(z+4))/((z+9)(z+4)))(((z+9)(z+6))/(z^2+4z))}}} Factor {{{z^2+15z+54}}} to get {{{(z+9)(z+6)}}}.



{{{(((z+4)(z+4))/((z+9)(z+4)))(((z+9)(z+6))/(z(z+4)))}}} Factor {{{z^2+4z}}} to get {{{z(z+4)}}}.



{{{((z+4)(z+4)(z+9)(z+6))/(z(z+9)(z+4)(z+4))}}} Combine the fractions. 



{{{(highlight((z+4))highlight((z+4))highlight((z+9))(z+6))/(z*highlight((z+9))highlight((z+4))highlight((z+4)))}}} Highlight the common terms. 

 

{{{(cross((z+4))cross((z+4))cross((z+9))(z+6))/(z*cross((z+9))cross((z+4))cross((z+4)))}}} Cancel out the common terms. 



{{{(z+6)/(z)}}} Simplify. 



So {{{((z^2+8z+16)/(z^2+13z+36))/((z^2+4z)/(z^2+15z+54))}}} simplifies to {{{(z+6)/(z)}}}.



In other words, {{{((z^2+8z+16)/(z^2+13z+36))/((z^2+4z)/(z^2+15z+54))=(z+6)/(z)}}} where {{{z<>-9}}}, {{{z<>-6}}}, {{{z<>-4}}}, or {{{z<>0}}}.