Ceiling Wiring for Recessed Lights

Hi, I'd like to tackle installing 6 recessed lights onto my kitchen ceiling. There is already a light fixture from which I will use to draw my power and I will be using retrofit/remodelling light kits. My question/fear is about running the wire across the ceiling joists. As I will have to cross several joists to run 2 parallel rows of lights (3 on each row), I'm wondering if anyone has any advice or tips on running the wire while minimizing cuts to the drywalled ceilings. Is cutting holes so I can drill the wiring through each joist inevitable? or are there any special tools/techniques others are familiar with.
BTW, I can't go in from the floor above since it is finished--and my skill level is a novice as a first time homeowner. Thank you in advance. Greatly appreciate any insights or experiences.
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Since this sounds like you first DIY project I'll actually give you some drywall advice instead, instead of trying to fish the electrical wire everwhere just cut a 1 foot wide strip the length of the kitchen out of the drywall. TRUST ME it's just as easy or easier to patch one big hole as itis to fix several smaller holes. If you cut the hole right down the centre it should leave room to run each wire down the sides.
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I could go either way on that. I used a large hole saw that I also used to make the cutouts for the recessed lights. I saved the circular cut out pieces as they were already textured correctly. After the first hole, I was able to predict where the remaining joists were.
When it came time to patch these access holes, I glued a flat piece of wood (a door shim) inside and across the hole. When that glue dried, I glued and screwed the circle of drywall back into the hole and patched with mud and popcorn later.
I had layers of blown in insulation, so a larger hole would have made a much larger mess.
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First, you want to be sure the circuit you're planning to use has enough capacity for what you're going to install. Once you've cut out your holes for the recessed fixtures, you may be able to minimize the sheetrock cuts by using a four foot extension with a spade bit, working through the recessed holes, but be careful not to drill through the floor up above, not to mention wires and pipes etc.

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there are electric auger bits for these applications 3 feet or longer with a loop on the business end of the bit, to attach and pull wire back if needed.,
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