Question 14

p = 2^70

To determine the value of p, we start with the expression d^d, which can be rewritten using the value of d. Given that d = 2⁶⁴, we find that d^d = (2⁶⁴)^(2⁶⁴) = 2^(64 * 2⁶⁴) = 2^(2^6 * 2⁶⁴) = 2^(2^(6 + 64)) = 2^70.

A) 128

Option A is incorrect because the expression 2^70 does not simplify to 2^128. The exponent of 128 does not relate to the multiplication of the exponents derived from d = 2⁶⁴.

B) 2⁶⁶

Option B is also incorrect as it does not match the derived exponent. The calculation shows that we reach 2^70, not 2⁶⁶, as the resulting exponent from d^d.

C) 2^70

This option is correct as it accurately represents the result of the exponentiation. By calculating d^d = (2⁶⁴)^(2⁶⁴), we arrive at an exponent of 2^70, confirming that p = 2^70.

D) 2^128

Option D is incorrect since 2^128 does not align with our calculations. The multiplication of the exponents leads to 2^70, not 2^128, highlighting a misunderstanding in exponent rules.

E) 2^384

Option E is incorrect as well. The exponent 2^384 does not correlate with the operations performed on d = 2⁶⁴, which results in 2^70, making this option irrelevant.

Conclusion

The correct answer is p = 2^70 because it directly results from the calculations based on the exponentiation of d = 2⁶⁴. All other options fail to match the derived value, demonstrating a misunderstanding of exponent multiplication and simplification. Thus, only option C accurately reflects the value of p in the given expression.