Consider the below figure wherein three circles touch each other externally.

Since the distances between the centres of these circles are 5 cm, 6 cm and 7 cm respectively, we have the following set of equations with respect to the above diagram:
x+y = 5     …..(1)

y+z = 6     ......(2) ($\Rightarrow$ y=6-z)...   (2.1)
x+z = 7     …..(3)

Adding (1), (2) and (3), we have $2(x+y+z)=5+6+7=18$
$⟹x+y+z=\mathrm{9....}\left(4\right)$
Using (1) in (4), we have $5+z=9⟹z=4$
Now using, $\left(3\right)⟹x=7-z=7-4=3$
And $\left(2.1\right)⟹y=6-z=6-4=2$  

Therefore, the radii of the circles are 3 cm, 2 cm and 4 cm.