You might think the answer is "very carefully", but in the case of Opuntia fragilis, it turns out to be "not at all".
Considering its aversion to reproduction, one wonders how the brittle prickly pear manages to continue the species? The fragility of its pads provides its survival mechanism. The pads break off easily, and this is actually the main way they move around. See: http://goo.gl/8WSCX