Solve the following simultaneous equations for $$a$$ and $$b$$<br>$$a\sqrt{a} + b\sqrt{b} = 183$$, $$a\sqrt{b} + b\sqrt{a} = 182$$

PLACEHOLDER ANSWER - the answer is a pair of fractions. With $$u = \sqrt{a}$$ and $$v = \sqrt{b}$$, the equations are $$u^3 + v^3 = 183$$ and $$uv(u + v) = 182$$, so $$(u + v)^3 = 183 + 3 \times 182 = 729$$ and $$u + v = 9$$, $$uv = \frac{182}{9}$$. Then $$u$$ and $$v$$ are the roots of $$9t^2 - 81t + 182 = 0$$, i.e. $$\frac{14}{3}$$ and $$\frac{13}{3}$$. So $$(a, b) = \left(\frac{196}{9}, \frac{169}{9}\right)$$ or $$\left(\frac{169}{9}, \frac{196}{9}\right)$$.