Which value among `2^(400), 3^(300) and 6^(200)` is the largest ?

Options

A. `6^(200)`
B. `3^(300)`
C. `2^(400)`
D. All are equal

Answer

To determine which value among \( 2^{400} \), \( 3^{300} \), and \( 6^{200} \) is the largest, we can rewrite each expression in a comparable form.

### Step 1: Rewrite each expression
We can express each of the powers in terms of a common exponent. Let's rewrite each term with a power of \( 100 \):

1. \( 2^{400} = (2^4)^{100} \)
2. \( 3^{300} = (3^3)^{100} \)
3. \( 6^{200} = (6^2)^{100} \)

### Step 2: Simplify the bases
Now we simplify the bases:

1. \( 2^4 = 16 \)
2. \( 3^3 = 27 \)
3. \( 6^2 = 36 \)

### Step 3: Compare the bases
Now we have:

- \( 2^{400} = 16^{100} \)
- \( 3^{300} = 27^{100} \)
- \( 6^{200} = 36^{100} \)

Since all three expressions are raised to the same power of \( 100 \), we can compare the bases directly:

- \( 16 \)
- \( 27 \)
- \( 36 \)

### Step 4: Determine the largest base
Among \( 16 \), \( 27 \), and \( 36 \), it is clear that:

- \( 36 > 27 > 16 \)

### Conclusion
Thus, \( 6^{200} = 36^{100} \) is the largest among the three values.

Therefore, the answer is \( 6^{200} \).

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