Simplify Expression 947De1

1. **State the problem:** Simplify the expression $$\left(\frac{a}{1-a}\right)^{1-a} + \left(\frac{1-a}{a}\right)^a.$$ 2. **Recall the formula and rules:** We want to simplify sums of terms with exponents involving $a$ and $1-a$. Notice the terms are powers of reciprocal bases with complementary exponents. 3. **Rewrite the expression:** Let $$x = \frac{a}{1-a}.$$ Then the expression becomes $$x^{1-a} + \left(\frac{1}{x}\right)^a = x^{1-a} + x^{-a}.$$ 4. **Rewrite powers:** $$x^{1-a} = x \cdot x^{-a}.$$ So the expression is $$x \cdot x^{-a} + x^{-a} = x^{-a}(x + 1).$$ 5. **Substitute back:** $$x = \frac{a}{1-a},$$ so $$x + 1 = \frac{a}{1-a} + 1 = \frac{a + (1-a)}{1-a} = \frac{1}{1-a}.$$ 6. **Final expression:** $$x^{-a}(x + 1) = \left(\frac{a}{1-a}\right)^{-a} \cdot \frac{1}{1-a} = \frac{1}{(1-a)} \cdot \left(\frac{a}{1-a}\right)^{-a}.$$ 7. **Rewrite the negative exponent:** $$\left(\frac{a}{1-a}\right)^{-a} = \left(\frac{1-a}{a}\right)^a.$$ 8. **Combine:** $$\frac{1}{1-a} \cdot \left(\frac{1-a}{a}\right)^a = \frac{1}{1-a} \cdot \frac{(1-a)^a}{a^a} = \frac{(1-a)^a}{(1-a) a^a} = \frac{(1-a)^{a-1}}{a^a}.$$ **Final simplified form:** $$\boxed{\frac{(1-a)^{a-1}}{a^a}}.$$