Consider the following statements:
(A) `a^0 =1 , a ne 0`
(R) `a^m + a^n = a^(m - n)`, m , n being integers, Of these statements :

Options

A. both A and R are true and R is the correct explanation of A
B. both A and R are true and R is the correct explanation of A
C. A is true , but R is false
D. A is false , but R is true

Answer

To solve the question, we need to analyze the two statements provided:

1. **Assertion (A)**: \( a^0 = 1 \) where \( a \neq 0 \)
2. **Reason (R)**: \( a^m + a^n = a^{m - n} \), where \( m \) and \( n \) are integers.

Let's evaluate each statement step by step.

### Step 1: Evaluate the Assertion (A)
The assertion states that \( a^0 = 1 \) for any non-zero \( a \).

- **Explanation**: According to the laws of exponents, any non-zero number raised to the power of zero is equal to one. This is a standard rule in mathematics.
- **Conclusion**: The assertion \( a^0 = 1 \) is **true** as long as \( a \neq 0 \).

### Step 2: Evaluate the Reason (R)
The reason states that \( a^m + a^n = a^{m - n} \).

- **Explanation**: This statement is not correct. The correct property of exponents states that \( a^m + a^n \) cannot be simplified to \( a^{m - n} \). Instead, it can be factored as \( a^n(a^{m-n} + 1) \) if \( n \leq m \) or \( a^m(1 + a^{n-m}) \) if \( m < n \). Thus, the equation \( a^m + a^n = a^{m - n} \) is false.
- **Conclusion**: The reason \( a^m + a^n = a^{m - n} \) is **false**.

### Final Conclusion
- The assertion (A) is true.
- The reason (R) is false.

### Answer Selection
Based on our evaluations:
- The correct option is: **3. A is true, but R is false.**