Find number of roots of the equation ` x^(3)-log_(0.5) x = 0`.

Explanation

To find the number of roots of the equation \( x^3 - \log_{0.5} x = 0 \), we can analyze the functions involved and their behavior.

### Step 1: Rewrite the equation
We can rewrite the equation as:
\[ x^3 = \log_{0.5} x \]

### Step 2: Understand the behavior of \( \log_{0.5} x \)
The logarithmic function \( \log_{0.5} x \) can be expressed using the change of base formula:
\[ \log_{0.5} x = \frac{\log x}{\log 0.5} \]
Since \( \log 0.5 < 0 \), the function \( \log_{0.5} x \) is a decreasing function. It approaches \( -\infty \) as \( x \) approaches \( 0^+ \) and approaches \( 0 \) as \( x \) approaches \( 1 \).

### Step 3: Analyze the function \( x^3 \)
The function \( y = x^3 \) is a cubic function that is increasing for all \( x \). It passes through the origin (0,0) and approaches \( +\infty \) as \( x \) increases.

### Step 4: Find intersections
To find the number of roots, we need to determine how many times the curve \( y = x^3 \) intersects the curve \( y = \log_{0.5} x \).

1. As \( x \) approaches \( 0^+ \), \( \log_{0.5} x \) approaches \( -\infty \) while \( x^3 \) approaches \( 0 \).
2. At \( x = 1 \), \( \log_{0.5} 1 = 0 \) and \( x^3 = 1 \).
3. For \( x > 1 \), \( \log_{0.5} x \) continues to decrease and approaches \( -\infty \) as \( x \) increases.

### Step 5: Determine the number of intersections
Since \( y = x^3 \) is increasing and \( y = \log_{0.5} x \) is decreasing, they can intersect at most once.

- At \( x = 1 \), we have \( x^3 = 1 \) and \( \log_{0.5} 1 = 0 \), which means \( x^3 > \log_{0.5} x \) at this point.
- As \( x \) approaches \( 0^+ \), \( x^3 < \log_{0.5} x \).

Thus, there is exactly one point where \( x^3 \) crosses \( \log_{0.5} x \).

### Conclusion
The number of roots of the equation \( x^3 - \log_{0.5} x = 0 \) is:
\[ \text{Number of roots} = 1 \]