+1172 Let
P = 3^{1/3} \cdot 9^{1/9} \cdot 27^{1/27} \cdot 81^{1/81}.
Then P can be expressed in the form \sqrt[a]{b}, where $a$ and $b$ are positive integers. Find the smallest possible value of $a + b.$
+130462 \(P = 3^{1/3} \cdot 9^{1/9} \cdot 27^{1/27} \cdot 81^{1/81} \)
\( \sqrt[a]{b} \)
P = 3^(1/3) * (3^2)^(1/9) * (3^3)^(1/27) * (3^4)^(1/81)
P = 3^(1/3) * 3^(2/9) * 3^(1/9) *3^(4/81)
P = 3 ^( 1/3 + 2/9 + 1/9 + 4/81)
P = 3^ [ ( 27 + 18 + 9 + 4 ) / 81 ]
P = 3 ^ [ 58 / 81] = 81 √ (3*58)
a + b = 81 + 3^58 = 81 + 4710128697246244834921603689 = 4710128697246244834921603770
