algebr

$\begin{aligned} a^3-b^3 & = (a-b)(a^2+ab+b^2) = 117 \\ 3(a^2+ab+b^2) & = 117 \\ a^2+ab+b^2 & = 39 \quad \quad ... (1) \end{aligned}$

$\begin{aligned} a-b & = 3 \\ (a-b)^2 & = 9 \\ a^2 - 2ab + b^2 & = 9 \quad \quad ... (2) \end{aligned}$

From $(1) - (2): \ \ 3ab = 30 \implies ab = 10$ , then

$\begin{aligned} (1): \ \ a^2+ab+b^2 & = 39 \\ a^2+2ab+b^2 & = 39 + ab \\ (a+b)^2 & = 39+10 = 49 \\ \implies a+b & = \pm 7 \end{aligned}$

$\implies$ the positive value of $a+b= \boxed{7}$ .