Which of the following is correct it `A = 3^(3^3^3) , B = 3^(33^3), C = 3^(3^33) and D = 3^(333)` ?

Options

A. a. `A gt B = C gt D`
B. b. `C gt A gt B gt D`
C. c. `A gt C gt D gt B`
D. d. `C gt B gt D gt A`

Answer

To determine which of the expressions \( A = 3^{3^{3^3}} \), \( B = 3^{33^3} \), \( C = 3^{3^{33}} \), and \( D = 3^{333} \) is the largest, we will evaluate each expression step by step.

### Step 1: Evaluate \( A = 3^{3^{3^3}} \)

First, we need to calculate the exponent \( 3^3 \):
\[ 3^3 = 27 \]
Now substitute this back into \( A \):
\[ A = 3^{3^{27}} \]

### Step 2: Evaluate \( B = 3^{33^3} \)

Next, we calculate \( 33^3 \):
\[ 33^3 = 33 \times 33 \times 33 = 35937 \]
Now substitute this back into \( B \):
\[ B = 3^{35937} \]

### Step 3: Evaluate \( C = 3^{3^{33}} \)

Now, we calculate \( 3^{33} \):
\[ 3^{33} \text{ is a large number, but we will keep it as is for comparison.} \]
So, we have:
\[ C = 3^{3^{33}} \]

### Step 4: Evaluate \( D = 3^{333} \)

Finally, we calculate \( D \):
\[ D = 3^{333} \]

### Step 5: Compare the values

Now we need to compare the exponents of each expression:

- For \( A = 3^{3^{27}} \)
- For \( B = 3^{35937} \)
- For \( C = 3^{3^{33}} \)
- For \( D = 3^{333} \)

#### Comparing \( A \) and \( B \):
- \( 3^{27} \) is much smaller than \( 35937 \), so \( A < B \).

#### Comparing \( B \) and \( C \):
- \( 3^{33} \) is larger than \( 35937 \) (since \( 3^{33} \) is approximately \( 5.2 \times 10^{15} \)), so \( C > B \).

#### Comparing \( C \) and \( D \):
- \( 3^{3^{33}} \) is much larger than \( 3^{333} \) since \( 3^{33} \) is a very large exponent.

### Conclusion:
From the comparisons:
- \( D > C > B > A \)

Thus, the largest expression is \( D = 3^{333} \).

### Final Answer:
The correct answer is \( D \).

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