To find the value of \( r^3 + s^3 + t^3 \) where \( r, s, t \) are the real roots of the equation
\[ (x - \sqrt[3]{13})(x - \sqrt[3]{53})(x - \sqrt[3]{103}) = 0, \]
we first identify the roots. The roots are given directly as:
\[ r = \sqrt[3]{13}, \quad s = \sqrt[3]{53}, \quad t = \sqrt[3]{103}. \]
We can compute \( r^3, s^3, \) and \( t^3 \) as follows:
\[ r^3 = 13, \quad s^3 = 53, \quad t^3 = 103. \]
Now we calculate \( r^3 + s^3 + t^3 \):
\[ r^3 + s^3 + t^3 = 13 + 53 + 103. \]
We perform the addition:
\[ 13 + 53 = 66, \] \[ 66 + 103 = 169. \]
Thus, we have:
\[ r^3 + s^3 + t^3 = 169. \]
Therefore, the value of \( r^3 + s^3 + t^3 \) is
\[ \boxed{169}. \]