The first thing to try would be to create custom crop ratios that meet those dimensions. Crop the image first, and then export copies only specifying either the long side or the short side. But even then the exact pixel dimensions might not be realized if the crop ratio is close to another "standard" crop ratio.
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I need to resize images to very specific dimensions. They are 550x440 and 2880x2304.
They are both exactly 1.25:1.00 or 8x10 Aspect Ratio.
Select that in the Crop tool 'Aspect' and center the image where you want it. As JimHess suggested specify 'Long Edge' in the Export module and enter the larger dimension (550 and 2880). This insures that both Landscape and Portrait orientation images will be properly resized to those exact dimensions (550x440 and 2880x2304).
I guess I should have calculated the ratios myself.
It's all good advice! The OP should be aware that if the desired custom Aspect Ratio is close (within two decimal places) to an existing Aspect ratio that's what LR will choose. This will cause the actual file dimensions to be slightly different. In this case both are exactly 1.25000000 so no issue.