Question

$x\sqrt{x}+y\sqrt{y}=183$
$x\sqrt{y}+y\sqrt{x}=182$

Answer 1

$\sqrt{y}=uory=u^2={\left(\frac{14}{3}\right)}^2=\frac{196}{9}$
$\sqrt{x}=torx=t^2={\left(\frac{13}{3}\right)}^2=\frac{169}{9}$
$u-t=\sqrt{{\left(u+t\right)}^2-4ut}$
$=\sqrt{{\left(9\right)}^2-4\times \frac{182}{9}}$
$=\sqrt{81-\frac{728}{9}}$
$=\sqrt{\frac{729-728}{9}}$
$=\sqrt{\frac{1}{9}}$
$u-t=\frac{1}{3}$
$u+t=9\left(adding\right)$
$------$
$2u=\frac{1}{3}+9$
$2u=\frac{28}{3}$
$u=\frac{14}{3}$
$also,u+t=9$
$t=9-u$
$=9-\frac{14}{3}$
$=\frac{27-14}{3}$
$t=\frac{13}{3}$
$x=\frac{169}{9},y=\frac{196}{9}$