If \( y \, \mathbin{\Diamond} \, x = y^{2x} \) for all positive integers, then \( (3 \, \mathbin{\Diamond} \, 4) \, \mathbin{\Diamond} \, 2 = \)
Answer (2)
The custom binary operation y \¤</u> x is defined as y times 2 raised to the power of x. By successively applying this operation, (3 \¤</u> 4) \¤</u> 2 evaluates to 192.
The question asks us to evaluate the expression (3 \¤</u> 4) \¤</u> 2 given that for all positive integers the operation defined by y \¤</u> x = y2x holds. This is a problem of function evaluation and the use of a custom binary operation. We shall apply this operation step-by-step.
First, evaluate 3 \¤</u> 4. According to the definition given, this means to multiply 3 by 24, or 24=16, so we get 3 × 16=48.
Then, we take the result 48 and apply the operation with 2, hence evaluating 48 \¤</u> 2. This means to multiply 48 by 22 or 22=4, so we get 48 × 4=192.
Therefore, the final answer is 192.
The expression ( 3 ◊ 4 ) ◊ 2 evaluates to 1853020188851841 by first calculating 3 ◊ 4 = 3 8 = 6561 and then 6561 ◊ 2 = 656 1 4 = 3 32 .
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