Fractionizers

$\begin{cases} 15^x = \dfrac 35 & \implies x \log 15 = \log 3 - \log 5 \implies x = \dfrac {\log 3 - \log 5}{\log 15} \\ y^{15} = \dfrac 35 & \implies y = \left(\dfrac 35\right)^\frac 1{15} \end{cases}$

$\implies x + y = \dfrac {\log 3 - \log 5}{\log 15} + \left(\dfrac 35\right)^\frac 1{15} \approx \boxed{0.78}$

Solving for $x$ :

$\begin{aligned} 15^{x} & = \dfrac{3}{5} \\ x & = \log_{15}{\dfrac{3}{5}} \\ x & = \dfrac{\log \dfrac{3}{5}}{\log 15} \end{aligned}$

Solving for $y$ :

$\begin{aligned} y^{15} & = \dfrac{3}{5} \\ \sqrt[15]{y^{15}} & = \sqrt[15]{\dfrac{3}{5}} \\ y & = \dfrac{\sqrt[15]{3}}{\sqrt[15]{5}} \end{aligned}$

Solving for $x+y$ :

$\dfrac{\log \dfrac{3}{5}}{\log 15} + \dfrac{\sqrt[15]{3}}{\sqrt[15]{5}} \approx -\dfrac{0.51086}{2.70805} + \dfrac{1.07599}{1.11326} \approx -0.18863+0.96652 \approx \boxed{0.78}$