Solve Exponent Aae7Af

1. **State the problem:** We are given two equations involving powers of 8 and 27 with the same exponent $x$: $$8^x + 27^x = \frac{13}{36}$$ $$8^x - 27^x = \frac{5}{36}$$ We need to find the value of $x$. 2. **Use substitution:** Let $$a = 8^x$$ $$b = 27^x$$ Then the system becomes: $$a + b = \frac{13}{36}$$ $$a - b = \frac{5}{36}$$ 3. **Solve for $a$ and $b$:** Add the two equations: $$a + b + a - b = \frac{13}{36} + \frac{5}{36}$$ $$2a = \frac{18}{36} = \frac{1}{2}$$ $$a = \frac{1}{4}$$ Subtract the second from the first: $$a + b - (a - b) = \frac{13}{36} - \frac{5}{36}$$ $$2b = \frac{8}{36} = \frac{2}{9}$$ $$b = \frac{1}{9}$$ 4. **Recall definitions:** $$a = 8^x = \frac{1}{4}$$ $$b = 27^x = \frac{1}{9}$$ 5. **Rewrite bases as powers:** $$8 = 2^3 \implies 8^x = (2^3)^x = 2^{3x}$$ $$27 = 3^3 \implies 27^x = (3^3)^x = 3^{3x}$$ So: $$2^{3x} = \frac{1}{4} = 2^{-2}$$ $$3^{3x} = \frac{1}{9} = 3^{-2}$$ 6. **Equate exponents:** From the first: $$3x = -2 \implies x = -\frac{2}{3}$$ From the second: $$3x = -2 \implies x = -\frac{2}{3}$$ 7. **Conclusion:** Both equations give the same value $$\boxed{x = -\frac{2}{3}}$$ which corresponds to option E.