Determine the number of real roots of the equation
${x^8 -x^7 + 2x^6- 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x + \frac{5}{2}= 0}$
1. **Rewrite the given equation:**
   \[
   x^8 - x^7 + 2x^6 - 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x + \frac{5}{2} = 0
   \]
   We can rewrite it as:
   \[
   x^8 - x^7 + 2x^6 - 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x = -\frac{5}{2}
   \]

2. **Factor the left-hand side (LHS):**
   Notice that the LHS can be factored as:
   \[
   x(x-1)(x^6 + 2x^4 + 3x^2 + 4)
   \]
   Therefore, the equation becomes:
   \[
   x(x-1)(x^6 + 2x^4 + 3x^2 + 4) = -\frac{5}{2}
   \]

3. **Analyze the sign of the LHS for \(x > 1\) or \(x < 0\):**
   - For \(x > 1\):
     - \(x > 0\)
     - \(x - 1 > 0\)
     - \(x^6 + 2x^4 + 3x^2 + 4 > 0\) (since all terms are positive)
     - Therefore, \(x(x-1)(x^6 + 2x^4 + 3x^2 + 4) > 0\)
   - For \(x < 0\):
     - \(x < 0\)
     - \(x - 1 < 0\)
     - \(x^6 + 2x^4 + 3x^2 + 4 > 0\) (since all terms are positive)
     - Therefore, \(x(x-1)(x^6 + 2x^4 + 3x^2 + 4) > 0\)

   Hence, there are no roots in \((-\infty, 0) \cup (1, +\infty)\).

4. **Analyze the interval \(0 \leq x \leq 1\):**
   - For \(0 \leq x \leq 1\):
     - \(x \geq 0\)
     - \(x - 1 \leq 0\)
     - \(x^6 + 2x^4 + 3x^2 + 4 \geq 4\) (since \(x^6, 2x^4, 3x^2 \geq 0\))
     - Therefore, \(x(x-1)(x^6 + 2x^4 + 3x^2 + 4) \leq 0\)

   To find the maximum absolute value of \(x(x-1)\):
   - The function \(f(x) = x(1-x)\) has a maximum at \(x = \frac{1}{2}\):
     \[
     f\left(\frac{1}{2}\right) = \frac{1}{2} \left(1 - \frac{1}{2}\right) = \frac{1}{4}
     \]

   To find the maximum value of \(x^6 + 2x^4 + 3x^2 + 4\) in \([0, 1]\):
   - The function \(g(x) = x^6 + 2x^4 + 3x^2 + 4\) is increasing in \([0, 1]\):
     \[
     g(1) = 1^6 + 2 \cdot 1^4 + 3 \cdot 1^2 + 4 = 10
     \]

   Therefore, the maximum value of \(|x(x-1)(x^6 + 2x^4 + 3x^2 + 4)|\) in \([0, 1]\) is:
   \[
   \left| \frac{1}{4} \cdot 10 \right| = \frac{10}{4} = \frac{5}{2}
   \]

   Since \(|x(x-1)(x^6 + 2x^4 + 3x^2 + 4)| < \frac{5}{2}\) for \(x \in [0, 1]\), there are no real roots in \([0, 1]\) either.

Conclusion:
There are no real roots for the given equation.

The final answer is \( \boxed{ 0 } \)