1. **State the problem:** Find the positive solution of the equation $$4x^{\frac{7}{8}} - 13 = 499$$. 2. **Isolate the term with the fractional exponent:** Add 13 to both sides: $$4x^{\frac{7}{8}} - 13 + 13 = 499 + 13$$ $$4x^{\frac{7}{8}} = 512$$ 3. **Divide both sides by 4 to solve for $x^{\frac{7}{8}}$:** $$\frac{4x^{\frac{7}{8}}}{\cancel{4}} = \frac{512}{\cancel{4}}$$ $$x^{\frac{7}{8}} = 128$$ 4. **Rewrite the equation to solve for $x$:** Recall that $x^{\frac{7}{8}} = (x^{\frac{1}{8}})^7$, so to isolate $x$, raise both sides to the reciprocal power $\frac{8}{7}$: $$\left(x^{\frac{7}{8}}\right)^{\frac{8}{7}} = 128^{\frac{8}{7}}$$ $$x = 128^{\frac{8}{7}}$$ 5. **Simplify $128^{\frac{8}{7}}$:** Note that $128 = 2^7$, so: $$x = \left(2^7\right)^{\frac{8}{7}} = 2^{7 \times \frac{8}{7}} = 2^8 = 256$$ 6. **Final answer:** $$\boxed{256}$$ is the positive solution to the equation.