To find the value of the given expression \({2^{{2^{{2^2}}}}}\), we need to evaluate the exponents from the top down. This type of expression, where exponents are stacked, is evaluated layer by layer, starting with the topmost exponent.

Evaluating the Exponents

Let's break down the calculation step-by-step:

1. First, evaluate the top exponent, which is \(2^2\).

   \(2^2 = 2 \times 2 = 4\)

2. Now, substitute this value back into the expression. The expression becomes \({2^{{2^{4}}}}\).

   The new topmost exponent is now \(2^4\).

3. Next, evaluate this exponent, \(2^4\).

   \(2^4 = 2 \times 2 \times 2 \times 2\)

   \(2^4 = 4 \times 4 = 16\)

4. Finally, substitute this result back into the expression. The expression is now \({2^{16}}\).

So, the value of \({2^{{2^{{2^2}}}}}\) is \({2^{16}}\).

Comparing with Options

Let's look at the given options and compare them with our calculated value, \({2^{16}}\).

• Option 1: \({2^6}\)
• Option 2: \({2^8}\)
• Option 3: \({2^{16}}\)
• Option 4: \({2^{222}}\)

Our calculated value \({2^{16}}\) matches Option 3.

Therefore, the value that is the same as \({2^{{2^{{2^2}}}}}\) is \({2^{16}}\).