1. **State the problem:** Given the equation $$2^{5x} \times 4^{9y} = \frac{1}{8}$$, show that $$5x + 2y = -3$$. 2. **Rewrite the bases:** Note that $$4 = 2^2$$ and $$\frac{1}{8} = 2^{-3}$$. 3. **Express the equation with base 2:** $$2^{5x} \times (2^2)^{9y} = 2^{-3}$$ 4. **Simplify the exponents:** $$2^{5x} \times 2^{18y} = 2^{-3}$$ 5. **Use the rule of multiplying powers with the same base:** $$2^{5x + 18y} = 2^{-3}$$ 6. **Since the bases are equal, set the exponents equal:** $$5x + 18y = -3$$ 7. **Rewrite the equation to isolate the form asked:** Divide both sides by 9: $$\frac{5x}{9} + 2y = -\frac{1}{3}$$ 8. **Multiply both sides by 9 to clear denominators:** $$\cancel{9} \times \left(\frac{5x}{\cancel{9}} + 2y\right) = \cancel{9} \times -\frac{1}{3}$$ $$5x + 18y = -3$$ 9. **Check the problem statement:** It asks to show $$5x + 2y = -3$$, but from the original equation, we have $$5x + 18y = -3$$. 10. **Conclusion:** The correct relation derived is $$5x + 18y = -3$$, not $$5x + 2y = -3$$. Possibly a typo in the problem statement. **Final answer:** $$5x + 18y = -3$$