Find the meaning of the expression $\sqrt[3]{19 - a} + \sqrt[3]{10 - a}$, if $\sqrt[3]{19 - a} - \sqrt[3]{10 - a} = 1$.
What i tried: $$\sqrt[3]{19 - a} = m $$ $$\sqrt[3]{10 - a} = n $$ $$ m - n = 1 $$ $$ m = 1 + n $$ $$ (1 + n) + n = 1 + 2n =$$ $$ ... $$
Also, $$m^3-n^3=\left(\sqrt[3]{19-a}\right)^3-\left(\sqrt[3]{10-a}\right)^3=19-a-(10-a)=9,$$ which gives $$m^2+mn+n^2=9$$ or $$1+3mn=9$$ or $$mn=\frac{8}{3},$$ which gives $$m^2+2mn+n^2=9+\frac{8}{3}$$ or $$(m+n)^2=\frac{35}{3},$$ which gives $$m+n=\sqrt{\frac{35}{3}}$$ or $$m+n=-\sqrt{\frac{35}{3}}.$$