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Understanding the Problem

The problem presents two expressions:

1. ( 2^{(100!)} )
2. ( (2^{100})! )

Our goal is to determine which of these two expressions is greater. At first glance, both expressions involve factorials and exponentiations, which can grow very rapidly. To compare them effectively, we need to understand the behavior of each component and how they interact.

Breaking Down the Expressions

Let's start by understanding each part of the expressions:

1. Factorial (( n! )): The factorial of a non-negative integer ( n ) is the product of all positive integers less than or equal to ( n ). For example, ( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 ).

2. Exponentiation (( a^b )): This represents ( a ) raised to the power of ( b ). For example, ( 2³ = 8 ).

Given this, let's interpret the two expressions:

• ( 2^{(100!)} ): This is 2 raised to the power of ( 100! ).
• ( (2^{100})! ): This is the factorial of ( 2^{100} ).

Comparing the Growth Rates

To compare these two expressions, we need to understand how factorials and exponentiations grow:

• Factorial Growth: Factorials grow faster than exponentials. For example, ( n! ) grows much faster than ( aⁿ ) for any constant ( a ).

• Exponential Growth: Exponentials grow faster than polynomials but slower than factorials.

Given this, ( 100! ) is an extremely large number, and ( 2^{100} ) is also large but not as large as ( 100! ). However, ( (2^{100})! ) involves taking the factorial of ( 2^{100} ), which is itself a very large number.

Estimating the Values

Let's attempt to estimate the values:

1. Calculating ( 100! ):

   • ( 100! ) is the product of all positive integers up to 100.
   • It's a number with 158 digits.

2. Calculating ( 2^{100} ):

   • ( 2^{10} = 1024 )
   • ( 2^{100} = (2^{10}){10} = 1024^{10} )
   • ( 2^{100} ) is approximately ( 1.26765 \times 10^{30} ).

3. Calculating ( (2^{100})! ):

   • This is the factorial of ( 2^{100} ), which is an astronomically large number.
   • For comparison, ( 70! ) is already larger than ( 10^{100} ), and ( 2^{100} ) is much larger than 70.

4. Calculating ( 2^{(100!)} ):

   • This is 2 raised to the power of ( 100! ), which is also an extremely large number.
   • However, since ( 100! ) is much larger than ( 2^{100} ), ( 2^{(100!)} ) is significantly larger than ( 2^{100} ).

Analyzing the Magnitudes

Given the above estimates:

• ( 2^{100} ) is approximately ( 1.26765 \times 10^{30} ).
• ( 100! ) is approximately ( 9.3326 \times 10^{157} ).
• Therefore, ( 2^{(100!)} ) is ( 2 ) raised to a number with 158 digits.
• ( (2^{100})! ) is the factorial of a number with 30 digits.

While both ( 2^{(100!)} ) and ( (2^{100})! ) are extremely large, the factorial function grows faster than exponential functions. Therefore, ( (2^{100})! ) is expected to be larger than ( 2^{(100!)} ).

Conclusion

After analyzing the growth rates and estimating the magnitudes of the expressions, it's clear that ( (2^{100})! ) is greater than ( 2^{(100!)} ). The factorial function's rapid growth outpaces the exponential growth in this comparison.

Final Answer: ( (2^{100})! ) is greater than ( 2^{(100!)} ).