12. For all positive integers a and b, the operation o× is defined by a o× b = 2^a + 2^b. If x, y, and w are positive integers, which of the following is equal to (x o× y) o× w?
Answer: A
(x o× y) o× w = 2^x + 2^y + 2^w
When performing the operation defined as a o× b = 2^a + 2^b, we first calculate x o× y, which results in 2^x + 2^y. Then applying the operation again with w gives us (2^x + 2^y) o× w = 2^(x o× y) + 2^w = (2^x + 2^y) + 2^w = 2^x + 2^y + 2^w.
A) 2^x + 2^y + 2^w
This option is correct because it accurately represents the result of the operation (x o× y) o× w. Following the operation's definition, we see that applying o× twice results in the sum of the powers of 2 from all three integers.
B) (2^x + 2^y) * (2^w)
This option is incorrect as it suggests multiplying the result of x o× y by 2^w instead of adding it. The operation o× is defined as a sum of powers of 2, not a multiplication.
C) 4^x + 4^y + 4^w
This option is also incorrect. While 4^x can be rewritten as (2^2)^x = 2^(2x), this does not follow the operational definition of o×. The operation results in a sum of powers of 2, not powers of 4.
D) 2^x + 2^y + 2^w
Although this option appears to be correct, it is misleading as it doesn't represent the correct notation for the operation performed twice. The correct result from the operation (x o× y) o× w is equivalent to the expression given in option A.
E) 2*((x^2 + y^2) + w^w)
This option is incorrect because it introduces polynomial terms and does not align with the definition of the operation o×. The operation should only involve powers of 2, not polynomial expressions.
Conclusion
The correct answer, A, accurately reflects the result of applying the operation o× twice as defined. All other options either misinterpret the operation's definition or introduce incorrect mathematical expressions, confirming that A is the only viable choice.