In aptitude (exponential equations simplification), if 4^(x + y) = 256 and 256^(x - y) = 4, where x and y are real numbers, determine the values of x and y using properties of powers of 4.

Introduction / Context:
This question involves solving a system of exponential equations by expressing all terms in the same base. Exponential equations of this type are common in algebra and aptitude exams, and the key trick is to rewrite 256 and 4 as powers of a common base, here base 4. Once the exponents are rewritten, the equations can be converted to linear equations in x and y and solved using standard algebraic techniques.

Given Data / Assumptions:
- 4^(x + y) = 256.
- 256^(x - y) = 4.
- x and y are real numbers.
- 256 and 4 can be expressed as powers of 4: 256 = 4^4 and 4 = 4^1.

Concept / Approach:
The concept is to convert both equations to the same base and then equate exponents. For the first equation, we directly equate x + y to the exponent that produces 256 when base is 4. For the second equation, we write 256 as 4^4, then use rules of exponents to combine powers, and finally equate the resulting exponent to 1 to match 4 = 4^1. This reduces the problem to solving a pair of linear equations in two variables x and y.

Step-by-Step Solution:
Step 1: Rewrite 256 in terms of base 4: 256 = 4^4.Step 2: From the first equation 4^(x + y) = 256, we get 4^(x + y) = 4^4, so x + y = 4.Step 3: For the second equation, 256^(x - y) = 4.Step 4: Substitute 256 = 4^4: (4^4)^(x - y) = 4.Step 5: Apply the power rule (a^b)^c = a^(b * c): 4^(4(x - y)) = 4.Step 6: Since 4 = 4^1, equate exponents: 4(x - y) = 1.Step 7: Therefore, x - y = 1 / 4.Step 8: Now we have a system of linear equations: x + y = 4 and x - y = 1 / 4.Step 9: Add the two equations: (x + y) + (x - y) = 4 + 1 / 4.Step 10: This gives 2x = 17 / 4, so x = (17 / 4) / 2 = 17 / 8.Step 11: Substitute x into x + y = 4: 17 / 8 + y = 4.Step 12: Write 4 as 32 / 8: 17 / 8 + y = 32 / 8, so y = (32 / 8) - (17 / 8) = 15 / 8.

Verification / Alternative check:
Check in the original equations. First equation: 4^(x + y) = 4^(17/8 + 15/8) = 4^(32/8) = 4^4 = 256, which is correct. Second equation: 256^(x - y) = 256^(17/8 - 15/8) = 256^(2/8) = 256^(1/4). Since 256 = 4^4, 256^(1/4) = (4^4)^(1/4) = 4, which matches the right-hand side. Thus, the pair (x, y) = (17/8, 15/8) satisfies both equations.

Why Other Options Are Wrong:
- 17/4, 15/4: These are exactly double the correct values, suggesting a mistake in dividing by 2 when solving for x and y from 2x and 2y equations.
- 9/17, 15/17 and 8/17, 8/15: These fractions do not satisfy x + y = 4 and x - y = 1/4 when tested, so they cannot be correct solutions of the system.
- 15/4, 17/4: This reverses the roles of x and y and uses wrong magnitudes; checking them fails in the original equations.

Common Pitfalls:
Common mistakes include confusing exponent rules, such as thinking (4^4)^(x - y) equals 4^(x - y) instead of 4^(4(x - y)), or incorrectly converting 256 to base 4. Some students also mis-handle fractions when solving the linear system, leading to values like 17/4 instead of 17/8. Working step by step and checking each algebraic transformation helps avoid these errors.

Final Answer:
The values of the variables are x = 17/8 and y = 15/8.