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I got a cheap MacBook Air that would only power on some of the time. I’ve noticed some corrosion where the MagSafe board attaches to the logic board. The MacBook Air won’t charge, but will start to boot and shut down again repeatedly. Can anyone guide me in what to about it?

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Can’t see much from the picture, but I trust your description. If you have corrosion around the connector it is likely a water damaged Mba and that’s the reason why you got it cheap. Most probably it got shorted by the spill and some power/charging components/power rail got burnt. It will require diagnostic and component level soldering skills..are you up for the task and have the necessary tools ?