My first thought is that this isn't powder but in fact wear on the laminate floor. After closer inspection, I'm not so sure about that as the spots of white at the top don't look like wear. Do you have any idea what this powder might have been? If it is some type of product lodged into the grain of the timber you'll need a brush to get it out. The bristles will be able to penetrate down into the grooves and hopefully remove it. I would dampen the surface down with some warm soapy water, let it sit for 5 minutes and then try scrubbing the white substance out with a toothbrush.
Please let us know if using a toothbrush and warm water has any effect on the substance.