If d = 2⁶⁴ and d^d = 2^p then p =

Rationale

p = 2^70.

To solve for p in the equation d^d = 2^p, we first recognize that d = 2^64. Therefore, d^d can be expressed as (2^64)^(2^64), which simplifies to 2^(64 * 2^64). This can be further simplified to 2^(2^70) since 64 * 2^64 equals 2^70. Thus, we conclude that p = 2^70.

A (128) YOUR ANSWER

This choice indicates a numerical value, but it does not reflect the exponential form necessary to describe p correctly. The calculations show that p must be expressed as an exponent of 2, specifically related to the powers derived from d's value, making this option incorrect.

B (2⁶⁶) YOUR ANSWER

While this option maintains the correct exponential format, it underestimates the actual value of p derived from the calculations. The expression for d^d leads to a much larger exponent than 66, thus making this choice incorrect.

C (2^70) CORRECT

This option correctly identifies p derived from the calculations performed. We established that d^d simplifies to 2^(64 * 2^64), which equals 2^(2^70). Therefore, this choice accurately represents the solution to the equation.

D (2^128) YOUR ANSWER

This choice suggests a much higher value for p than can be justified by the calculations. Since d^d simplifies to 2^(2^70), it is evident that p cannot equal 2^128, rendering this option incorrect.

E (2^384) YOUR ANSWER

This option significantly exceeds the required value for p. There is no calculation that supports 2^384 as a valid outcome from the expression d^d, solidifying this choice as incorrect.

Conclusion

The problem centers on finding the value of p in the equation d^d = 2^p, where d = 2^64. Through simplification, we determined that p equals 2^70, making option C the correct answer. Other choices either miscalculate or misinterpret the exponential relationships at play, emphasizing the importance of precise mathematical manipulation in deriving solutions.

If d = 2⁶⁴ and d^d = 2^p then p =

p = 2^70.

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