I haven't used Nair "Power" but I've used regular Nair for Men from my local CVS. It does work on thick hair (I'm Greek and have thick dark hair). You just leave it on for 8 minutes and wipe the hair away. However, you MUST take a shower right after to wash the hair-free area with soap and water. If you don't you will experience uncomfortable skin irritation. As long as you immediately wash the skin with soap and water, you will not have even the slightest discomfort.
Disclaimer: I don't use this stuff for cycling, but instead use it for a certain area that isn't all that accessible for shaving. I wouldn't use this stuff all over my body. I only use it for that one hard to shave place and then quickly shower afterwards. It's too potent/acidic/toxic smelling for me to want to use it all over my body. If I cared enough to remove the hair from my legs, I would definitely use a razor and shaving cream.