If xy= 16, find minimum value of x+y

Answer

To find the minimum value of \( x + y \) given that \( xy = 16 \), we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-Step Solution:

1. **Understand the Problem**: We are given that \( xy = 16 \) and need to find the minimum value of \( x + y \).

2. **Apply the AM-GM Inequality**: According to the AM-GM inequality, for any two non-negative numbers \( x \) and \( y \):
\[ \frac{x + y}{2} \geq \sqrt{xy} \]
This means that the average of \( x \) and \( y \) is at least as large as the square root of their product.

3. **Substitute the Given Value**: Since we know \( xy = 16 \), we can substitute this into the inequality:
\[ \frac{x + y}{2} \geq \sqrt{16} \]

4. **Calculate the Square Root**: The square root of 16 is 4:
\[ \frac{x + y}{2} \geq 4 \]

5. **Multiply Both Sides by 2**: To eliminate the fraction, multiply both sides by 2:
\[ x + y \geq 8 \]

6. **Conclusion**: The minimum value of \( x + y \) is 8, which occurs when \( x = y \). Since \( xy = 16 \), we can find \( x \) and \( y \) by solving the equations:
\[ x = y \quad \text{and} \quad xy = 16 \implies x^2 = 16 \implies x = 4, y = 4 \]
Thus, \( x + y = 4 + 4 = 8 \).

### Final Answer:
The minimum value of \( x + y \) is **8**.