Question 1209817
Let's break down the problem step by step:

1. **Express all terms with the same base:**
   * P = 3^(1/3) * (3^2)^(1/9) * (3^3)^(1/27) * (3^4)^(1/81)
   * P = 3^(1/3) * 3^(2/9) * 3^(3/27) * 3^(4/81)

2. **Combine the exponents:**
   * P = 3^(1/3 + 2/9 + 3/27 + 4/81)

3. **Find a common denominator:**
   * The least common denominator is 81.
   * P = 3^(27/81 + 18/81 + 9/81 + 4/81)

4. **Add the fractions:**
   * P = 3^(58/81)

5. **Express in radical form:**
   * P = (3^58)^(1/81)
   * P = ⁸¹√(3^58)

6. **Identify a and b:**
   * a = 81
   * b = 3^58

7. **Calculate a + b:**
   * a + b = 81 + 3^58

8. **Calculate 3^58:**
   * 3^58 = 4710128697246244834921603770

9. **Calculate a+b**
   * a + b = 81 + 4710128697246244834921603770 = 47101286972462448349216037751

Therefore, the smallest possible value of a + b is 81 + 3^58.