Which one of the following is greatest one?

Options

A. a. `3^(3322)`
B. b. `33^(322)`
C. c. `333^(22)`
D. d. `22^(333)`

Answer

To determine which of the given options is the greatest, we will compare the values systematically based on their powers and bases. Let's denote the options as follows:

1. \( a = 3^{3^{2^2}} \)
2. \( b = 22^{2^{3^3}} \)
3. \( c = 33^{3^{2^2}} \)
4. \( d = 33^{3^{3^2}} \)

We will compare these options step by step.

### Step 1: Simplify the expressions
First, we will simplify the expressions to make comparisons easier.

1. **For \( a = 3^{3^{2^2}} \)**:
- Calculate \( 2^2 = 4 \)
- So, \( a = 3^{3^4} = 3^{81} \)

2. **For \( b = 22^{2^{3^3}} \)**:
- Calculate \( 3^3 = 27 \)
- So, \( b = 22^{2^{27}} \)

3. **For \( c = 33^{3^{2^2}} \)**:
- Again, \( 2^2 = 4 \)
- So, \( c = 33^{3^4} = 33^{81} \)

4. **For \( d = 33^{3^{3^2}} \)**:
- Calculate \( 3^2 = 9 \)
- So, \( d = 33^{3^9} = 33^{19683} \)

### Step 2: Compare the values
Now we will compare the values of \( a \), \( b \), \( c \), and \( d \):

- **Compare \( a \) and \( c \)**:
- \( a = 3^{81} \)
- \( c = 33^{81} \)
- Since \( 33 > 3 \), \( c > a \).

- **Compare \( c \) and \( d \)**:
- \( c = 33^{81} \)
- \( d = 33^{19683} \)
- Since \( 19683 > 81 \), \( d > c \).

- **Compare \( b \) with \( c \)**:
- \( b = 22^{2^{27}} \)
- To compare \( b \) with \( c \), we need to evaluate \( 22^{2^{27}} \) and \( 33^{81} \).
- Since \( 2^{27} \) is a very large exponent, we can estimate that \( b \) is likely larger than \( c \) because \( 22 \) raised to a very high power can exceed \( 33^{81} \).

### Conclusion
From the comparisons:
- \( d = 33^{19683} \) is the largest.
- \( b = 22^{2^{27}} \) is likely larger than \( c = 33^{81} \).
- \( a = 3^{81} \) is the smallest.

Thus, the greatest one among the options is:

**Answer: \( d = 33^{3^{3^2}} \)**

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