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Lift off the top bar of the vanity light bar. You should now see copper wiring (covered in black plastic and white plastic). Feel the copper ends of the cord you split. Take the one that feels smooth and twist it together with the black vanity cord. Take the one that feels rough and twist it together with the white vanity cord. You can simply twist the cords by crossing them over repeatedly like they're twist ties. Make sure the wire nuts for both the white and black wires in the light bar are affixed firmly. Set the cover back onto the vanity light bar so that all the cords are covered.When doing this, try to keep the white and black wires separate from each other within the light bar. Separating the cords can prevent the cords from touching, which can cause the circuit breaker to trip, or even cause a fire. Take the metallic caps that you set aside earlier and set them back into place around the sides of each electrical socket. Screw one light bulb into each electrical socket so that it's firmly in place. Plug the extension cords into power outlets and turn on your vanity lights. If you're using a wireless remote control, you may need to follow the manufacturer's instructions regarding setup. Mark the back of the mirror where you'd like to screw in hardware.  Some mirrors already come with hooks attached to the back, so you simply need to hang the mirror on the wall. Measure the distance between the hooks. Measure the same distance on the wall where you're hanging the mirror and make a small mark. Use a drill to attach hardware (like a wall anchor or screw) to the wall so you can hang your mirror. Make sure you weigh the mirror and use hardware that supports that weight. Otherwise, your makeup mirror may damage your wall or fall down.
Connect the extension cord to the vanity light. Attach the light bar cover. Screw in the light bulbs. Hang the mirror.