Solving the Exponential Power Tower $2^{2^{2^2}}$

The question asks to find the value equivalent to the power tower expression $2^{2^{2^2}}$. We evaluate this expression starting from the top exponent downwards.

Step-by-Step Evaluation

1. Evaluate the top exponent: First, calculate $2^2$. $2^2 = 4$
2. Substitute and evaluate the next exponent: Substitute the result back into the expression. It becomes $2^{2^4}$. Now, calculate $2^4$. $2^4 = 2 \times 2 \times 2 \times 2 = 16$
3. Final Result: Substitute this result back. The expression simplifies to $2^{16}$. $2^{2^{2^2}} = 2^{16}$

Comparing with Options

The calculated value is $2^{16}$. Comparing this with the given options:

• Option 1: $2^6$
• Option 2: $2^8$
• Option 3: $2^{16}$
• Option 4: $2^{222}$

The expression $2^{2^{2^2}}$ is equivalent to $2^{16}$, which corresponds to Option 3.