Number of straight lines represented by `x^5+y^5 = 0 ` is ……

Answer

To determine the number of straight lines represented by the equation \( x^5 + y^5 = 0 \), we can follow these steps:

### Step 1: Rearranging the Equation
We start with the equation:
\[ x^5 + y^5 = 0 \]
We can rearrange this to isolate one variable:
\[ x^5 = -y^5 \]

### Step 2: Expressing in Terms of a Ratio
Next, we can express the equation in terms of a ratio:
\[ \frac{x^5}{y^5} = -1 \]
This implies:
\[ \left(\frac{x}{y}\right)^5 = -1 \]

### Step 3: Finding the Roots
The equation \( \left(\frac{x}{y}\right)^5 = -1 \) suggests that we are looking for the fifth roots of \(-1\). The fifth roots of \(-1\) can be expressed in polar form:
\[ \frac{x}{y} = e^{i\left(\frac{\pi}{5} + \frac{2k\pi}{5}\right)} \quad \text{for } k = 0, 1, 2, 3, 4 \]
This results in five complex solutions.

### Step 4: Identifying Real Solutions
However, we are interested in real solutions. The only real fifth root of \(-1\) is:
\[ \frac{x}{y} = -1 \]
This corresponds to the line:
\[ x + y = 0 \quad \text{or} \quad y = -x \]

### Step 5: Conclusion
Since there is only one real solution, we conclude that the number of straight lines represented by the equation \( x^5 + y^5 = 0 \) is:
\[ \text{1} \]

### Final Answer
The number of straight lines represented by \( x^5 + y^5 = 0 \) is **1**.

---