My daughter doesn't have insurance and is experiencing toothache with back molar. which is most cost effective, root canal or extraction?

Cost effective: The most cost effective treatment is the one that is the least invasive and which preserves the dentition. If a tooth has root canal treatment it may need a crown. If extracted it may need to be replaced with implant/crown. Not knowing which tooth is involved and it's functional position in your daughter's dentition, the best advice I can give you is seek a quality dental examination.