$$\left(a^{\frac{1}{z - x}}\right)^{\frac{1}{z - y}}.\left(a^{\frac{1}{x - y}}\right)^{\frac{1}{x - z}}.\left(a^{\frac{1}{y - z}}\right)^{\frac{1}{y - x}} =$$

$$\left(a^{\frac{1}{z - x}}\right)^{\frac{1}{z - y}}.\left(a^{\frac{1}{x - y}}\right)^{\frac{1}{x - z}}.\left(a^{\frac{1}{y - z}}\right)^{\frac{1}{y - x}} =$$

$$=a^{\left(\frac{1}{\left(x-z\right)\left(y-z\right)}\right)}\cdot a^{\left(\frac{1}{\left(x-y\right)\left(x-z\right)}\right)}\cdot a^{\left(-\frac{1}{\left(y-z\right)\left(x-y\right)}\right)}$$

$$=a^{\left(\frac{1}{\left(x-z\right)\left(y-z\right)}+\frac{1}{\left(x-y\right)\left(x-z\right)}-\frac{1}{\left(y-z\right)\left(x-y\right)}\right)}$$

Adding up all the terms in the powers yields 0 as the answer. 
Hence, a raised to 0 = 1