When ` x^(m) ` is multiplied by ` x^(n) ` product is 1 . The relation between m and n is

Options

A. m n = 1
B. m = n
C. m + n = 1
D. m = - n

Answer

To solve the problem, we need to find the relationship between \( m \) and \( n \) when \( x^m \) is multiplied by \( x^n \) and the product equals 1.

### Step-by-step Solution:

1. **Understanding the Multiplication of Exponents**:
When we multiply two powers with the same base, we add their exponents. Therefore, we can express the multiplication as:
\[ x^m \cdot x^n = x^{m+n} \]

2. **Setting the Product Equal to 1**:
According to the problem, the product of \( x^m \) and \( x^n \) is equal to 1:
\[ x^{m+n} = 1 \]

3. **Using the Property of Exponents**:
We know that any non-zero number raised to the power of 0 equals 1. Therefore, for the equation \( x^{m+n} = 1 \) to hold true, the exponent must be 0:
\[ m+n = 0 \]

4. **Finding the Relationship Between \( m \) and \( n \)**:
From the equation \( m+n = 0 \), we can isolate \( m \):
\[ m = -n \]

### Conclusion:
The relationship between \( m \) and \( n \) is:
\[ m = -n \]