Which of the following is greatest number?

Options

A. `3^(2^(3))`
B. `3^(4^(2))`
C. `3^(3^(2))`
D. `3^(2^(2))`

Answer

To determine which of the following numbers is the greatest, we need to evaluate each option step by step. The options are:

- Option A: \(3^{2^3}\)
- Option B: \(3^{4^2}\)
- Option C: \(3^{3^2}\)
- Option D: \(3^{2^2}\)

### Step 1: Evaluate each exponentiation

1. **Option A: \(3^{2^3}\)**
- Calculate \(2^3 = 8\)
- So, \(3^{2^3} = 3^8\)

2. **Option B: \(3^{4^2}\)**
- Calculate \(4^2 = 16\)
- So, \(3^{4^2} = 3^{16}\)

3. **Option C: \(3^{3^2}\)**
- Calculate \(3^2 = 9\)
- So, \(3^{3^2} = 3^9\)

4. **Option D: \(3^{2^2}\)**
- Calculate \(2^2 = 4\)
- So, \(3^{2^2} = 3^4\)

### Step 2: Compare the results

Now we have:
- Option A: \(3^8\)
- Option B: \(3^{16}\)
- Option C: \(3^9\)
- Option D: \(3^4\)

### Step 3: Identify the greatest exponent

The exponents we need to compare are:
- \(8\) (from \(3^8\))
- \(16\) (from \(3^{16}\))
- \(9\) (from \(3^9\))
- \(4\) (from \(3^4\))

Clearly, \(16\) is the greatest exponent.

### Conclusion

Thus, the greatest number among the options is **Option B: \(3^{4^2}\) or \(3^{16}\)**.

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