Problem Statement:
Two circles with radii 10cm and 8cm intersect, and the length of their common chord is 12cm. We need to find the distance between their centers.

Solution:
To find the distance between the centers of two intersecting circles, we can use the Pythagorean theorem and the properties of similar triangles.

Step 1: Draw the diagram:
Let's start by drawing a diagram to visualize the problem. Draw two circles with radii 10cm and 8cm, and label their centers as A and B. Draw a common chord between the two circles and label its length as 12cm. Also, label the points where the common chord intersects the circles as C and D.

Step 2: Identify the relevant lengths:
From the given information, we have:
- Radius of circle with center A = 10cm
- Radius of circle with center B = 8cm
- Length of common chord CD = 12cm

Step 3: Observe the right triangles:
In the diagram, we can observe two right triangles: triangle ABC and triangle BCD. Let's focus on triangle ABC first.

Step 4: Use the Pythagorean theorem:
In triangle ABC, we can apply the Pythagorean theorem to find the length of side AC, which is the distance between the centers of the circles.

Using the Pythagorean theorem, we have:
(AC)^2 = (AB)^2 + (BC)^2

Substituting the values, we get:
(AC)^2 = (10cm)^2 + (8cm)^2
(AC)^2 = 100cm^2 + 64cm^2
(AC)^2 = 164cm^2

Taking the square root of both sides, we find:
AC ≈ 12.81cm

Step 5: Calculate the distance between the centers:
Therefore, the distance between the centers of the two circles is approximately 12.81cm.

Conclusion:
The distance between the centers of two circles with radii 10cm and 8cm, where their common chord has a length of 12cm, is approximately 12.81cm.