Let m be the minimum possible value of _ 3 ( 3 ^ y _ 1 + 3 ^ y _ 2...

Applying A.M-G.M inequality,

$\left(\dfrac{{3}^{{y}_{1}} + {3}^{{y}_{2}} + {3}^{{y}_{3}}}{3}\right) \ge \left({3}^{{y}_{1}} ⋅ {3}^{{y}_{2}} ⋅ {3}^{{y}_{3}}\right)\dfrac{1}{3} = \left({3}^{{y}_{1} + {y}_{2} + {y}_{3}}\right)\dfrac{1}{3}$

$⇒ {3}^{{y}_{1}} + {3}^{{y}_{2}} + {3}^{{y}_{3}} \ge 81$

So, $m = \log_{3}\left(81\right) = 4$

Now, $\log_{3}{x}_{1} + \log_{3}{x}_{2} + \log_{3}{x}_{3} = \log_{3}\left({x}_{1} . {x}_{2} . {x}_{3}\right)$

$∵ \dfrac{{x}_{1} + {x}_{2} + {x}_{3}}{3} \ge \left({x}_{1} . {x}_{2} . {x}_{3}\right)\dfrac{1}{3} ⇒ {x}_{1} . {x}_{2} . {x}_{3} \le 27$ (applying A.M-G.M inequality)

So, $M = \log_{3}27 = 3$

Thus, $\log_{2}\left({m}^{3}\right) + \log_{3}\left({M}^{2}\right) = 8$