Indices

Concept:The expression simplifies using the property of exponents and common denominators.Explanation:Write each term by rewriting denominator as $1 + z^{a-b} + z^{a-c}$. Multiply numerator and denominator of first term by $z^b z^c$:$\frac{1}{1+z^{a-b}+z^{a-c}} = \frac{z^b z^c}{z^b z^c + z^a z^c + z^a z^b}$.Similarly, second term becomes $\frac{z^a z^c}{z^a z^c + z^b z^a + z^b z^c}$ and third term becomes $\frac{z^a z^b}{z^a z^b + z^b z^c + z^a z^c}$.All denominators are the same: $z^b z^c + z^a z^c + z^a z^b$.Summing the numerators gives $z^b z^c + z^a z^c + z^a z^b$.Thus the sum is $\frac{z^b z^c + z^a z^c + z^a z^b}{z^b z^c + z^a z^c + z^a z^b} = 1$.Answer:The expression reduces to $1$, which corresponds to option C.