Indices with chained definitions: If a^x = b, b^y = c and x*y*z = 1, find the value of c^z in terms of a, b, c.

Introduction / Context:
This checks your comfort with composing exponents when variables are defined by earlier exponent relations. The key is to rewrite everything in a single base, then use the product of exponents condition xyz = 1.

Given Data / Assumptions:

• a^x = b
• b^y = c
• x*y*z = 1

Concept / Approach:
Express b and c in terms of a. Then raise c to the power z and simplify the exponent using xyz = 1. This turns c^z into a familiar power of a.

Step-by-Step Solution:
b = a^xc = b^y = (a^x)^y = a^(xy)c^z = (a^(xy))^z = a^(xyz)Given xyz = 1 ⇒ c^z = a^1 = a

Verification / Alternative check:
Pick sample values x = 1, y = 1, z = 1 (so xyz = 1). Then b = a, c = a, and c^z = a^1 = a, as expected.

Why Other Options Are Wrong:
b or c would require xyz to equal 1/x or 1/(xy); ab or a/b do not follow from the exponent composition.

Common Pitfalls:
Forgetting that (a^m)^n = a^(mn). Avoid adding exponents here—multiplication is the rule.

Final Answer:
a