Finding Common Tangent of two circles

Hint:

Consider the height and the hypotenuse of the right triangle.

Consider the triangle $MNO$ with $O$ being the orthogonal projection of $N$ on the segment $[MC]$.

The length of $MC$ is $8$ centimeters. The length of $CO$ is $5$ centimeters.

Therefore the length of $OM$ is $3$ centimeters. Also, you know the length $$MN = MA + AN = 8 + 5$$

Now with the hypotenuse law, you can find that \begin{align}MN^2 &= MO^2 + ON^2 \\ 3^2 + x^2 &= (8+5)^2\end{align}