If ${\log }_2\mathrm{x}+{\log }_{\mathrm{x}}2=\frac{10}{3}={\log }_2\mathrm{y}+{\log }_{\mathrm{y}}2$  and $\mathrm{x}\ne \mathrm{y}$ , then $\mathrm{x}+\mathrm{y}=$

$\frac{10}{3}=3+\frac{1}{3}$ . The given equation is of the for

$\begin{array}{l}\mathrm{p}+\frac{1}{\mathrm{p}}=3+\frac{1}{3}=\mathrm{q}+\frac{1}{\mathrm{q}},\text{ where }\mathrm{p}\ne \mathrm{q}\text{ as }\mathrm{x}\ne \mathrm{y}\\ \Rightarrow {\log }_2\mathrm{x}=3,{\log }_2\mathrm{y}=1/3\\ \Rightarrow \mathrm{x}=2^3,\mathrm{y}=2^{1/3}\\ \therefore \mathrm{x}+\mathrm{y}=8+2^{1/3}\end{array}$