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In digging fence post holes in my yard I discovered there is a large tree root, approximately 3'' in diameter, right where I need a post. The hole is about 15' away from the trunk of the tree. Is it okay to sever the root where I need to dig?
That's a toughy without examining the situation first hand. Depending on factors like the size, age of the tree this can cause troubles for the tree.
Can you set a post in a differnt location?
If no, seek the advice of a tree expert.
I've cut many roots in my life, even larger than 3" in diameter, but each situation is different.
What kind of tree?
As the others have noted, the size and type of tree will make a difference, however the fact that you're 15' away from the tree and have a 3" root would indicate that the tree is well established and would have many other 3" feeder roots at its disposal, so the shortening of this one root isn't going to be a detriment to the tree. It is also important to note that you aren't removing the root, it is still 15' long and it will regrow from where it was cut, similarly to how branches regrow.
If you determine that cutting off this root will not endanger the tree, use a chain saw for a clean and easy cut. But before you crank your saw, completely remove the dirt around the root. Don't saw the dirt, it will dull your chain in 1 second.