Which of the following is /are correct?
`(i) a^(x + y) = ax + ay " " (ii) (a^x)^(y) = ya^(x)`
`(iii) a(x cdot y) = axcdotay " " (iv) (a^x)/(a^y) = a^(x - y)`

Options

A. (i) only
B. (ii) & (iii) only
C. (iv) only
D. None of these

Answer

To determine which of the given statements are correct, let's analyze each one step by step.

### Statement (i): \( a^{(x + y)} = ax + ay \)

1. **Understanding the statement**: This statement suggests that the exponentiation of a base \( a \) raised to the sum of \( x \) and \( y \) is equal to the sum of \( ax \) and \( ay \).
2. **Correct exponentiation rule**: The correct rule for exponents states that \( a^{(x + y)} = a^x \cdot a^y \).
3. **Conclusion**: Therefore, this statement is **incorrect**.

### Statement (ii): \( (a^x)^y = ya^x \)

1. **Understanding the statement**: This statement suggests that raising \( a^x \) to the power of \( y \) results in \( ya^x \).
2. **Correct exponentiation rule**: The correct rule states that \( (a^x)^y = a^{(x \cdot y)} \).
3. **Conclusion**: Therefore, this statement is **incorrect**.

### Statement (iii): \( a^{(x \cdot y)} = ax \cdot ay \)

1. **Understanding the statement**: This statement suggests that raising \( a \) to the product of \( x \) and \( y \) is equal to multiplying \( ax \) and \( ay \).
2. **Correct exponentiation rule**: The correct interpretation is \( a^{(x \cdot y)} \) does not equal \( ax \cdot ay \). Instead, \( ax \cdot ay = a^x \cdot a^y \).
3. **Conclusion**: Therefore, this statement is **incorrect**.

### Statement (iv): \( \frac{a^x}{a^y} = a^{(x - y)} \)

1. **Understanding the statement**: This statement suggests that dividing \( a^x \) by \( a^y \) results in \( a \) raised to the power of \( (x - y) \).
2. **Correct exponentiation rule**: The correct rule states that \( \frac{a^x}{a^y} = a^{(x - y)} \).
3. **Conclusion**: Therefore, this statement is **correct**.

### Final Conclusion:
- **Correct Statements**: Only statement (iv) is correct.
- **Incorrect Statements**: Statements (i), (ii), and (iii) are incorrect.