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?

Rationale

(x otimes y) otimes w = 2^x + 2^y + 2^w.

To solve the expression (x otimes y) otimes w, we first apply the operation defined by a otimes b = 2^a + 2^b. Therefore, we calculate x otimes y, which gives us 2^x + 2^y, and then we apply the operation again with w, resulting in 2^(2^x + 2^y) + 2^w, which simplifies to 2^x + 2^y + 2^w.

A (2^x + 2^y + 2^w) CORRECT

This choice correctly reflects the final result of the operation as outlined above. It accurately combines the results of the nested operations, confirming that (x otimes y) otimes w indeed equals 2^x + 2^y + 2^w.

B ((2^x + 2^y) * (2^w)) YOUR ANSWER

This expression would imply a multiplication of the results of the first operation with the third term, which is not how the defined operation otimes works. The otimes operation is additive, not multiplicative, so this choice does not reflect the correct computation.

C (4^x + 4^y + 4^w) YOUR ANSWER

This choice misrepresents the nature of the operation. While 4^a can be rewritten as (2^2)^a = 2^(2a), it does not correctly represent the additive results of the otimes operation applied in sequence, thus making it an incorrect representation.

D (2^x + 2^y + 2^w) YOUR ANSWER

This option is similar to A but incorrectly assumes a different form of the expression. It does not account for the correct application of the otimes operation in the sequence provided, leading to confusion.

E (2*((x^2 + y^2) + w^w)) YOUR ANSWER

This option introduces a completely different operation involving squares and exponentiation of w, which has no relationship to the otimes operation as defined. It strays far from the correct interpretation of the given operations.

Conclusion

The operation defined by a otimes b = 2^a + 2^b leads to the conclusion that (x otimes y) otimes w simplifies to 2^x + 2^y + 2^w, confirming that option A is correct. The other choices fail to adhere to the defined operation or misinterpret the resulting expressions, highlighting the importance of understanding operation definitions in mathematical contexts.

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?

(x otimes y) otimes w = 2^x + 2^y + 2^w.

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