Which is greater of the two : `2^(300)` or `3^(200)` ?

Answer

To determine which is greater between \(2^{300}\) and \(3^{200}\), we can follow these steps:

### Step 1: Rewrite the exponents to have a common base
We want to compare \(2^{300}\) and \(3^{200}\). To do this, we can express both numbers with a common exponent. We can rewrite \(2^{300}\) as \((2^3)^{100}\) and \(3^{200}\) as \((3^2)^{100}\).

### Step 2: Calculate the bases
Now we calculate the bases:
- \(2^3 = 8\)
- \(3^2 = 9\)

### Step 3: Rewrite the expressions
Now we can rewrite our expressions:
- \(2^{300} = (2^3)^{100} = 8^{100}\)
- \(3^{200} = (3^2)^{100} = 9^{100}\)

### Step 4: Compare the bases
Now we compare \(8^{100}\) and \(9^{100}\). Since both expressions are raised to the same power (100), we only need to compare the bases:
- \(8 < 9\)

### Step 5: Conclusion
Since \(8 < 9\), it follows that:
\[ 8^{100} < 9^{100} \]
Thus, we conclude that:
\[ 2^{300} < 3^{200} \]
Therefore, \(3^{200}\) is greater than \(2^{300}\).

### Final Answer
The greater value is \(3^{200}\).

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