If `a=2, b= 3` find the values of
`(a^(b) + b^(a))^(-1)` (ii) `(a^(a) + b^(b))^(-1)`

Answer

To solve the given problem step by step, we will evaluate two expressions based on the values of \( a = 2 \) and \( b = 3 \).

### Part (i): Find the value of \( (a^b + b^a)^{-1} \)

1. **Substitute the values of \( a \) and \( b \)**:
\[ a = 2, \quad b = 3 \]

2. **Calculate \( a^b \)**:
\[ a^b = 2^3 = 2 \times 2 \times 2 = 8 \]

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

4. **Add the results of \( a^b \) and \( b^a \)**:
\[ a^b + b^a = 8 + 9 = 17 \]

5. **Take the reciprocal**:
\[ (a^b + b^a)^{-1} = \frac{1}{17} \]

### Part (ii): Find the value of \( (a^a + b^b)^{-1} \)

1. **Calculate \( a^a \)**:
\[ a^a = 2^2 = 2 \times 2 = 4 \]

2. **Calculate \( b^b \)**:
\[ b^b = 3^3 = 3 \times 3 \times 3 = 27 \]

3. **Add the results of \( a^a \) and \( b^b \)**:
\[ a^a + b^b = 4 + 27 = 31 \]

4. **Take the reciprocal**:
\[ (a^a + b^b)^{-1} = \frac{1}{31} \]

### Final Answers:
- For part (i): \( (a^b + b^a)^{-1} = \frac{1}{17} \)
- For part (ii): \( (a^a + b^b)^{-1} = \frac{1}{31} \)

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