Time To #Log

Using $\log_{a^{2}}A = \frac{1}{2} \log_{a}A$ , Three equations can be written as $\\ \log_{2}x\sqrt{yz} =2 \\ \implies \log_{2}x^{2}yz = 4 \\ \log_{3}y\sqrt{zx} =2 \\ \implies \log_{3}y^{2}zx = 4 \\ \log_{4}z\sqrt{xy} =2 \\ \implies \log_{4}{z}^{2}xy = 4 \\ \implies x^{2}yz = 2^{4} , y^{2}zx = 3^{4} , z=^{2}xy = 4^{4} \\$ Multiplying all these equations,we get $\\ (xyz)^{4}=(2\times 3\times4)^{4} \\ \implies xyz=24 \\ Thus, (2\cdot 3\cdot 4)x = 4^2\Longleftrightarrow x = \frac {2}{3}, \\$ Analogously , from other equations , we obtain the answer ${x = \frac {2}{3},\ y = \frac {27}{8},\ z = \frac {32}{3}} \\$ which gives answer as $2+3+27+8+32+3 = \boxed {75}$