In algebra with exponents, let x and y be non zero real numbers such that (x / y)^(a − 4) = (y / x)^(2a − 5) for all real values of the parameter a for which these expressions are defined. Based on this condition, what can be concluded about the relationship between x and y?

Introduction / Context:
This question uses properties of exponential expressions to deduce a relationship between two non zero real numbers x and y. Rather than asking you to find specific values of x or y, it asks for a qualitative relationship. The key is to interpret the equality of powers carefully and recognise how it must hold for all real values of the parameter a, which imposes strong restrictions on the base x / y.

Given Data / Assumptions:
- x and y are non zero real numbers.
- The equality (x / y)^(a − 4) = (y / x)^(2a − 5) holds for all real values of a for which both sides are defined.
- We must determine the relationship between x and y implied by this identity.
- Standard rules of exponents and logarithms apply, and we consider real valued expressions.

Concept / Approach:
First, notice that y / x is the reciprocal of x / y. We can therefore rewrite the right hand side in terms of the same base as the left. This leads to an equation of the form (x / y)^(a − 4) = (x / y)^(something involving a). For this identity to hold for all admissible real values of a, either the exponents must match for all a, or the base x / y must be a special value such that any exponent produces the same result. Examining these possibilities reveals that the only consistent choice is x / y = 1, which implies x = y.

Step-by-Step Solution:
1) Start with (x / y)^(a − 4) = (y / x)^(2a − 5). 2) Observing that y / x = (x / y)^(−1), rewrite the right side as ((x / y)^(−1))^(2a − 5) = (x / y)^(−(2a − 5)). 3) The equation becomes (x / y)^(a − 4) = (x / y)^(−(2a − 5)). 4) Let k = x / y. Then the equation is k^(a − 4) = k^(−(2a − 5)). 5) For this to hold for all real a where the expressions are defined, the only way is for the base k to be such that k^t is the same for all exponents t. This happens only when k = 1 (for real, positive bases), since 1 raised to any power equals 1. 6) Therefore, x / y = 1, which implies x = y.

Verification / Alternative check:
If x = y, then x / y = 1 and y / x = 1. In that case, both sides of the original equation become 1 raised to some exponent, which is always 1 whenever the exponent is defined. Hence the equality holds for all real a, confirming that x = y is sufficient. Conversely, if x and y were unequal, then x / y would not equal 1, and k^t would vary with t for real exponents, making it impossible for k^(a − 4) to equal k^(−(2a − 5)) for all choices of a. Thus x = y is also necessary.

Why Other Options Are Wrong:
Options A (x > y), C (x < y), and E (x ≥ y) describe partial order relationships that do not guarantee the equality for all values of a. For most unequal positive x and y, the equality could hold at best for a few specific values of a, not for all. Option B (Cannot be determined) ignores the very strong condition that the equality is valid for every admissible a, which forces a unique relationship. Only Option D correctly captures the necessary and sufficient condition x = y.

Common Pitfalls:
A common misunderstanding is to equate the exponents directly for all a, which leads to a condition on a instead of on x and y, or to consider only a single value of a. Another pitfall is not recognising that 1 is the only positive real base for which raising to any power gives the same result. Thinking carefully about the behaviour of exponential functions with different bases is crucial to avoid these errors.

Final Answer:
The only way for the equality of powers to hold for all real values of a is if x / y = 1, that is, x = y.