, 2012

Expanding, $(a+b)^2=a^2+2ab+b^2=(a^2+b^2)+2ab$. Since $a^2+b^2=24$ and $ab=6$, then $(a+b)^2=24+2(6)=36$. Expanding, $(x+y)^2=x^2+2xy+y^2=(x^2+y^2)+2xy$. Since $(x+y)^2=13$ and $x^2+y^2=7$, then $13=7+2xy$ or $2xy=6$, and so $xy=3$. Expanding, $(j+k)^2=j^2+2jk+k^2=(j^2+k^2)+2jk$. Since $j+k=6$ and $j^2+k^2=52$, then $6^2=52+2jk$ or $2jk=-16$, and so $jk=-8$. Expanding, $(m^2+n^2)^2=m^4+2m^2n^2+n^4=(m^4+n^4)+2m^2n^2$. Since $m^2+n^2=12$ and $m^4+n^4=136$, then $12^2=136+2m^2n^2$ or $2m^2n^2=8$ or $m^2n^2=4$, and so $mn=\pm2$.