Which of the two is larger : `3^(12)` or `6^(6)`?

Answer

To determine which of the two numbers is larger, \(3^{12}\) or \(6^{6}\), we can follow these steps:

### Step 1: Rewrite \(6^6\) in terms of base 3
We know that \(6\) can be expressed as \(3 \times 2\). Therefore, we can rewrite \(6^6\) as:
\[ 6^6 = (3 \times 2)^6 \]
Using the property of exponents \((a \times b)^n = a^n \times b^n\), we can expand this:
\[ 6^6 = 3^6 \times 2^6 \]

### Step 2: Rewrite \(3^{12}\)
Next, we can express \(3^{12}\) in a way that allows us to compare it directly with \(6^6\):
\[ 3^{12} = 3^6 \times 3^6 \]

### Step 3: Compare the two expressions
Now we have:
- \(3^{12} = 3^6 \times 3^6\)
- \(6^6 = 3^6 \times 2^6\)

We can factor out \(3^6\) from both expressions:
\[ 3^{12} = 3^6 \times 3^6 \]
\[ 6^6 = 3^6 \times 2^6 \]

### Step 4: Simplify the comparison
Now we can compare \(3^6\) with \(2^6\):
\[ 3^{12} = 3^6 \times 3^6 \quad \text{and} \quad 6^6 = 3^6 \times 2^6 \]

To compare \(3^6\) and \(2^6\), we can calculate:
\[ 2^6 = 64 \quad \text{and} \quad 3^6 = 729 \]

### Step 5: Conclusion
Since \(729 > 64\), we can conclude that:
\[ 3^{12} > 6^6 \]
Thus, \(3^{12}\) is larger than \(6^{6}\).

### Final Answer
Therefore, \(3^{12}\) is larger than \(6^{6}\).

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