**Solution**

To find the value of \(a^3 b^3\), we need to determine the values of \(a\) and \(b\) first.

Given that \(a\) and \(b\) are the solutions of the quadratic equation \(x^2 - mx + m = 0\), we can use the quadratic formula to find their values.

**Finding the Solutions**

The quadratic formula states that for any quadratic equation of the form \(ax^2 + bx + c = 0\), the solutions can be found using the formula:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

In this case, the equation is \(x^2 - mx + m = 0\), so \(a = 1\), \(b = -m\), and \(c = m\). Substituting these values into the formula, we have:

\[x = \frac{-(-m) \pm \sqrt{(-m)^2 - 4(1)(m)}}{2(1)}\]

Simplifying further:

\[x = \frac{m \pm \sqrt{m^2 - 4m}}{2}\]

**Determining the Values of a and b**

Since \(a\) and \(b\) are the solutions of the quadratic equation, we can equate the quadratic formula to \(a\) and \(b\) respectively:

\[a = \frac{m + \sqrt{m^2 - 4m}}{2}\]
\[b = \frac{m - \sqrt{m^2 - 4m}}{2}\]

**Calculating a³ and b³**

To find the values of \(a^3\) and \(b^3\), we can cube the equations for \(a\) and \(b\) respectively:

\[a^3 = \left(\frac{m + \sqrt{m^2 - 4m}}{2}\right)^3\]
\[b^3 = \left(\frac{m - \sqrt{m^2 - 4m}}{2}\right)^3\]

Simplifying these expressions, we have:

\[a^3 = \frac{(m + \sqrt{m^2 - 4m})^3}{8}\]
\[b^3 = \frac{(m - \sqrt{m^2 - 4m})^3}{8}\]

**Calculating a³ b³**

Finally, we can find the value of \(a^3 b^3\) by multiplying \(a^3\) and \(b^3\):

\[a^3 b^3 = \frac{(m + \sqrt{m^2 - 4m})^3}{8} \times \frac{(m - \sqrt{m^2 - 4m})^3}{8}\]

Simplifying further, we have:

\[a^3 b^3 = \frac{(m + \sqrt{m^2 - 4m})(m - \sqrt{m^2 - 4m})^3}{64}\]

This expression represents the value of \(a^3 b^3\) in terms of \(m\).