Question

\[ (a^{x+1}/a^{y+1})^{x+y}(a^{y+2}/a^{z+2})^{y+z}(a^{z+3}/a^{x+3})^{z+x} \]

Solution

\[ = (a^{x-y})^{x+y}(a^{y-z})^{y+z}(a^{z-x})^{z+x} \] \[ = a^{(x-y)(x+y)} \cdot a^{(y-z)(y+z)} \cdot a^{(z-x)(z+x)} \] \[ = a^{(x^2-y^2)+(y^2-z^2)+(z^2-x^2)} \] \[ = a^0 \] \[ = 1 \]

Answer

\[ \boxed{1} \]

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