State True or False: `a^(b)gtb^(a)` is true. If a=3 and b=4, but false if a=2 and b=3.

Answer

To determine whether the statement \( a^b > b^a \) is true or false for the given values of \( a \) and \( b \), we will evaluate it step by step.

### Step 1: Evaluate for \( a = 3 \) and \( b = 4 \)

We need to check if \( 3^4 > 4^3 \).

1. Calculate \( 3^4 \):
\[ 3^4 = 3 \times 3 \times 3 \times 3 \]
\[ = 9 \times 3 = 27 \quad \text{(first two 3s)} \]
\[ = 27 \times 3 = 81 \quad \text{(adding the last 3)} \]

2. Calculate \( 4^3 \):
\[ 4^3 = 4 \times 4 \times 4 \]
\[ = 16 \times 4 = 64 \]

3. Compare the results:
\[ 81 > 64 \]
Thus, \( 3^4 > 4^3 \) is **true**.

### Step 2: Evaluate for \( a = 2 \) and \( b = 3 \)

Now, we need to check if \( 2^3 > 3^2 \).

1. Calculate \( 2^3 \):
\[ 2^3 = 2 \times 2 \times 2 = 8 \]

2. Calculate \( 3^2 \):
\[ 3^2 = 3 \times 3 = 9 \]

3. Compare the results:
\[ 8 > 9 \]
Thus, \( 2^3 > 3^2 \) is **false**.

### Conclusion

From our evaluations:
- For \( a = 3 \) and \( b = 4 \), the statement \( a^b > b^a \) is **true**.
- For \( a = 2 \) and \( b = 3 \), the statement \( a^b > b^a \) is **false**.

Therefore, the overall statement is **TRUE**.

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