Step-by-step Solution:

Let the radii of the two circles be \(r_1\) and \(r_2\).

1. Formulate equations from the given information:
The sum of the areas is given as \(130\pi\) sq. cm.
The area of a circle is \(\pi r^2\).
\[\pi r_1^2 + \pi r_2^2 = 130\pi\]
Dividing the entire equation by \(\pi\), we get our first equation:
\[r_1^2 + r_2^2 = 130 \quad \cdots(1)\]
Since the two circles touch externally, the distance between their centers is the sum of their radii.
\[r_1 + r_2 = 14 \quad \cdots(2)\]
2. Solve the system of equations:
We can use the algebraic identity \((a+b)^2 = a^2+b^2+2ab\).
Substitute the values from equations (1) and (2) into this identity:
\[\begin{aligned} (r_1 + r_2)^2 &= (r_1^2 + r_2^2) + 2r_1r_2 \\ (14)^2 &= 130 + 2r_1r_2 \\ 196 &= 130 + 2r_1r_2 \\ 196 - 130 &= 2r_1r_2 \\ 66 &= 2r_1r_2 \\ r_1r_2 &= 33 \end{aligned}\]
3. Find the radii by forming a quadratic equation:
We now know the sum of the radii (\(r_1+r_2=14\)) and the product of the radii (\(r_1r_2=33\)). The radii are the roots of the quadratic equation \(x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0\).
\[x^2 - 14x + 33 = 0\]
Factoring the equation:
\[(x - 11)(x - 3) = 0\]
The roots are \(x=11\) and \(x=3\).

Result:
The radii of the two circles are 11 cm and 3 cm.