If $$a^{b}b^{a} (c^{d} + d^{c}) = c^{d}d^{c} (a^{b} + b^{a}),$$ where $$a, b, c, d
eq 0,$$ which of the following statements is/are correct?

1. $$\frac{1}{b^{a}} + \frac{1}{a^{b}} = \frac{1}{c^{d}} + \frac{1}{d^{c}}$$

2. If $$a^{b} - d^{c} = 0,$$ then $$c^{d}$$ must be equal to $$b^{a},$$ for all real values of $$a, b, c, d.$$

3. Given $$c=0,$$ then $$a^{b} + b^{a} = 1,$$ assuming $$d$$ is a non-zero number.

Question

If $$a^{b}b^{a} (c^{d} + d^{c}) = c^{d}d^{c} (a^{b} + b^{a}),$$ where $$a, b, c, d 
eq 0,$$ which of the following statements is/are correct?

1. $$\frac{1}{b^{a}} + \frac{1}{a^{b}} = \frac{1}{c^{d}} + \frac{1}{d^{c}}$$

2. If $$a^{b} - d^{c} = 0,$$ then $$c^{d}$$ must be equal to $$b^{a},$$ for all real values of $$a, b, c, d.$$

3. Given $$c=0,$$ then $$a^{b} + b^{a} = 1,$$ assuming $$d$$ is a non-zero number.

Accepted Answer

**Solution:**

1. Starting with the given equation:
   $$a^{b}b^{a} (c^{d} + d^{c}) = c^{d}d^{c} (a^{b} + b^{a})$$

2. Rearranging terms, divide both sides by $$a^{b}b^{a}c^{d}d^{c}$$ (all non-zero):
   $$\frac{c^{d} + d^{c}}{c^{d}d^{c}} = \frac{a^{b} + b^{a}}{a^{b}b^{a}}$$

3. This simplifies to:
   $$\frac{1}{d^{c}} + \frac{1}{c^{d}} = \frac{1}{b^{a}} + \frac{1}{a^{b}}$$

4. Notice that the left side is $$\frac{1}{d^{c}} + \frac{1}{c^{d}}$$ and the right side is $$\frac{1}{b^{a}} + \frac{1}{a^{b}}$$, which is the reverse of statement 1. Statement 1 claims equality but swaps the sides incorrectly.

5. For statement 2, if $$a^{b} = d^{c}$$, substituting back into the original equation does not necessarily imply $$c^{d} = b^{a}$$ for all real values; this is not generally true.

6. Statement 3 assumes $$c=0$$, but since $$c^{d}$$ and $$d^{c}$$ appear in the original equation and $$c=0$$ would make terms like $$c^{d}$$ undefined or zero depending on $$d$$, the equation does not hold as stated.

7. Therefore, none of the statements 1, 2, or 3 are correct based on the given equation and conditions.

**Answer:** None of the above

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