**Solution:**

Let's assume that the roots of the quadratic equation ax^2 - 8x + c = 0 are equal and denoted by r.

Given that a * b * c = 9, we can write:

a * r^2 - 8 * r + c = 0 ...(1)

**Finding the value of a:**

To find the value of a, we need to find the coefficient of x^2 in the quadratic equation.

From equation (1), we can see that the coefficient of x^2 is a.

Therefore, a = a.

**Finding the value of c:**

To find the value of c, we need to find the constant term in the quadratic equation.

From equation (1), we can see that the constant term is c.

Therefore, c = c.

**Finding the value of b:**

To find the value of b, we can use the fact that the sum of the roots of a quadratic equation is given by:

Sum of roots = - (coefficient of x) / (coefficient of x^2)

In this case, the sum of roots is 2r (since the roots are equal).

Substituting the values in the equation, we get:

2r = - (-8) / a

2r = 8 / a

r = 4 / a

**Substituting the values:**

Now, let's substitute the values of a, b, and c in the equation a * b * c = 9 and solve for the values.

a * b * c = 9

a * (4/a) * c = 9

4c = 9

c = 9/4

Therefore, the value of the root r is 4/a, and substituting the value of c, we get:

r = 4/a = 4/(9/4) = 16/9

Hence, the value of the root is 16/9.

In summary:

- The coefficient of x^2, a, is equal to the original value of a.
- The constant term, c, is equal to the original value of c.
- The value of b is determined using the sum of roots formula, which gives us b = 8/a.
- Substituting the values of a, b, and c in the equation a * b * c = 9, we find the value of c.
- Finally, substituting the value of c in the expression for the root, we find the value of the root.

Answer - 8