Given information:
- Two circles of radii 10 cm and 8 cm intersect.
- The length of the common chord is 12 cm.

To find:
The distance between their centers.

Using the properties of circles and the given information, we can solve this problem step by step:

Step 1: Draw a diagram
- Draw two circles with radii 10 cm and 8 cm intersecting each other.
- Draw a common chord of length 12 cm between the circles.
- Label the centers of the circles as O1 and O2.

Step 2: Identify the right triangle
- Draw radii from the center of each circle to the endpoints of the common chord.
- This will form a right triangle with the common chord as the hypotenuse.

Step 3: Find the length of the perpendicular from the center of one circle to the common chord
- In the right triangle, the perpendicular from the center of one circle to the common chord acts as the altitude.
- Let this perpendicular be h.
- The length of the perpendicular can be found using the formula: h = (2 * r1 * r2) / d, where r1 and r2 are the radii of the circles and d is the length of the common chord.
- Substituting the given values, we get: h = (2 * 10 * 8) / 12 = 16/3 cm.

Step 4: Find the distance between the centers of the circles
- The distance between the centers of the circles can be found using the Pythagorean theorem.
- The hypotenuse of the right triangle formed in Step 2 is the sum of the radii of the two circles.
- Let the distance between the centers of the circles be x.
- Using the Pythagorean theorem, we have: x^2 = (r1 + r2)^2 - h^2.
- Substituting the given values, we get: x^2 = (10 + 8)^2 - (16/3)^2 = 18^2 - (16/3)^2 = 324 - 256/9 = 2576/9.
- Taking the square root of both sides, we get: x = √(2576/9) = √(2576)/√9 = 56/3 cm.

Therefore, the distance between the centers of the circles is 56/3 cm or approximately 13.3 cm. Thus, the correct answer is option D.