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Gents,
Using the following tools can I hope to make a professional looking Hope chest:
10" miter saw
Router and table and fence
Planer
Jig saw
Portable belt sander
Table saw
 

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Theo
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Yes. IF you have the skills.
 
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You can probably do the vast majority of this with your table saw, but you must tune it up properly before you start. That means making sure the miter slots, blade and fence are all aligned, with the fenc about 4/1000ths away from the blade to make sure you don't run the risk of kickback. Also, just go get or order a Wixey digital angle gauge. This little $30 gadget lets you set the blade angle precisely to 90 or 45 degrees to the table. This is critical. There are plenty of videos on YouTube on setting up your table saw.

There are many corner joints you can make that are easy to make on a properly set up saw. But nothing will work right if you don't spend an hour or two to tune up that saw. Also, use the Wixey on your miter saw so that the blade is exactly 90 to the base. If it's not exactly right, you will not be able to glue up anything you cut on it. Also, you need a really good right angle triangle or engineer's square (L shaped) Most cheapo right angle carpenter's or framing squares are not perfectly accurate. My bes framing square is off by nearly 2 degrees, enough to spoil any project.

My favorite square is a thick plastic Rockler draftsmans' triangle About $16. https://www.rockler.com/woodworker-s-triangles

The Wixey is about $30 https://www.amazon.com/Wixey-WR300-...457&sr=8-3&keywords=wixey+digital+angle+gauge

An engineer's square
set of 4 (I have this) is about $30 https://www.amazon.com/Woodstock-D4...=1546056569&sr=8-3&keywords=engineer's+square.

Here are some videos you might find helpful:

Here's a good video on setting up a table saw, one of MANY you will find on youtube

Here is a decent video on making basic cuts safely.

A locking rabbet cut can be great for assembling a drawer. Here's a video:

Here's more on cutting rabbets for building cabinets and some boxes.

Finally, I've attached a pdf of the 17 (plus) things that helped me accelerate my learning curve on woodworking. It's long, but has pictures. It may save you some costly mistakes or buying unnecessary tools.
 

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@DesertRatTom Don't forget the clamps the long ones,it will take at least 6 or 8 and the glue.

Gadzilla be sure to take pictures and post them as you go and ask lots of questions too. If you fill out your profile it will give us an idea how to answer your questions as related to your tools available and past experience.
Herb
 

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Thanks Herb. There are some quite nice clamps in larger sizes at Harbor Freight. I know a lot of us have used them. I'm particularly fond of the long aluminum bar clamps. Not terribly expensive as long clamps go.

Thinking about this a bit more, I also suggest he get some long open time glue. The regular stuff locks up a couple of minutes, barely time to line things up.
 

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Just realized I forgot to attach the pdf of 17 things. It is fairly long, and has pictures, so it's worth printing out. It covers a period of about 6 years when I went from DIY to woodworking. It may save you some missteps and money.

The video on rabbets that shows you hand planes is one way to make great rabbets, but if you're using plywood (probably are), then you might not find the hand plane works for you. It also shows using a Dado stack, Several blades stacked together to cut the rabbet. A good 8 inch dado set is expensive.

Instead, you can make a jig that fits over your fence to hold the workpiece vertically, which allows you to make a rabbet in two passes, once with the piece held vertically that cuts the depth of the rabbet, then a second cut with the piece laying flat, which cuts the width of the rabbet. This allows you to cut a rabbet that just fits your side panels.

Although I have a dado set, most often I use this second method. Attached is a picture of a vertical fence. Pretty simple to make, but again, take the time to set up and tune up your saw first. The jig's accuracy depends on exact 90 degree cuts (Wixey does it). Notice in the picture that there are two Rockler brand clamps holding the inner layer of the tall fence tight to the saw's fence. ($20 for 2) https://www.rockler.com/universal-fence-clamps

The second thing to notice is the material. It is Baltic Birch plywood, something you can get from a real wood supplier. It must be flat to start with. It is 17 layers thick for the 3/4 thickness (actually 18mm), but all those layers grain runs in different directions each layer, so it is extremely stable. It also has a beautifully finished side. The third thing to notice is the blue "T Track" that allows you to add on accessories. Not necessary at first. You have to drill the holes vertically, best done with a drill press, but you can also get drill guides, an inexpensive way to get a vertical hole with a hand drill. If you don't have a drill, you can find them cheap used, even new, with a power cord. Battery operated drills are much more expensive.

Here's a video with a lot of informaiton about jigs and cutting rabbets on the table saw. You will soon find that making jigs is not only fun, but dramatically increase the safety and versatility of your table saw. Notice that there is a "carrier" jig that helps you push your vertical workpiece through safely and with more precision than trying to hold it steady with your hands. No matter what your jigs, for the most part you want to make them of Baltic Birch. Flatness counts.
 

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So far, I've just emphasized using the table saw for rabbets, but you can also cut these with your router using a mortising bit. You can get rabbeting bits, but most are limited in how wide the will make the rabbet. The mortising bit can be purchased in 3/4 inch size. You adjust the width by moving the fence back, away from the front of the bit. Mortising bits are made to give you a flat bottomed cut. Make the rabbet in multiple passes, not more than 1/8th removed per pass. Ply has glue holding it together, and that really can build up heat in your bit if you take off too much, and that will destroy the temper, and your bit will be ruined.

Whiteside makes a great rabbeting bit that has bearings, but will cut a much wider rabbet than most. About $53 on Amazon. All positive reviews and a really good manufacturer. https://www.amazon.com/Whiteside-Ro...ds=3/4+inch+rabbet+router+bit#customerReviews

Here's the 3/4 inch mortising bit for about $30 on Amazon, you can also find this at most Home Depot stores. Freud bits are extremely popular because they're very good. https://www.amazon.com/Freud-Dia-Mo...sr=1-4&keywords=3/4+inch+router+mortising+bit

If you elect to use the table saw, you might consider replacing the stock blade with a full kerf "Glue Line" blade by Freud. It is one of their industrial blades and doesn't deflect, while giving an edge finish that rarely needs sanding or scraping. About $53 at Amazon https://www.amazon.com/Freud-Indust...sr=1-1&keywords=Freud+10+inch+glue+line+blade It's a rip blade but I use it for nearly everything. It also has one tooth in 4 that has a flat top so it makes a smooth bottom cut.
 
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