Sum of Real Values of Exponential Equation

Express the base terms as powers of $2$: $\frac{1}{8} = 2^{-3}$ and $\frac{1}{32768} = 2^{-15}$.

Substitute into the equation: $(2^{-3})^k \times (2^{-15})^{1/3} = 2^{-3} \times (2^{-15})^{1/k}$ $2^{-3k - 5} = 2^{-3 - \frac{15}{k}}$

Equating exponents: $-3k - 5 = -3 - \frac{15}{k}$ $3k + 2 = \frac{15}{k}$ $3k^2 + 2k - 15 = 0$

For a quadratic equation $3k^2 + 2k - 15 = 0$, the discriminant $D = 4 - 4(3)(-15) = 184 > 0$, so both roots are real. Sum of real roots = $-\frac{b}{a} = -\frac{2}{3}$.