To find the value of \( a^3 + b^3 \) where \( a \) and \( b \) are the roots of the quadratic equation \( x^2 - 4x + 1 = 0 \), we can utilize the properties of roots and the formula for the sum of cubes.

Step 1: Identify the Roots
- The roots \( a \) and \( b \) can be found using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
- For our equation, \( a = 1, b = -4, c = 1 \):
\[
x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} = \frac{4 \pm \sqrt{16 - 4}}{2} = \frac{4 \pm \sqrt{12}}{2} = \frac{4 \pm 2\sqrt{3}}{2} = 2 \pm \sqrt{3}
\]
- Thus, the roots are \( a = 2 + \sqrt{3} \) and \( b = 2 - \sqrt{3} \).

Step 2: Calculate \( a + b \) and \( ab \)
- Using Vieta’s formulas:
- \( a + b = 4 \)
- \( ab = 1 \)

Step 3: Use the Sum of Cubes Formula
- The formula for the sum of cubes is:
\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]
- To find \( a^2 + b^2 \), use the identity:
\[
a^2 + b^2 = (a + b)^2 - 2ab = 4^2 - 2 \cdot 1 = 16 - 2 = 14
\]
- Therefore:
\[
a^2 - ab + b^2 = a^2 + b^2 - ab = 14 - 1 = 13
\]

Step 4: Calculate \( a^3 + b^3 \)
- Plugging into the sum of cubes formula:
\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2) = 4 \cdot 13 = 52
\]

Final Result
- The value of \( a^3 + b^3 \) is \( \boxed{52} \).