Screenshots of PDFs are going to be limited by the density of your monitor and (I'd assume) rendering settings from the PDF viewer.
There are many tools for converting a PDF file (or pages from a PDF file) to a PNG file. Adobe Acrobat is able to do this. Imagemagick, which is free, is also able to do this.
With Imagemagick installed, the command below extracts page 11 of foo.pdf as a PNG file at 300 DPI.
convert -density 300 foo.pdf foo.png
Note: the page number is in the square brackets. The page numbering starts at zero, so page 1 is
Once you have the page as a PNG, use a traditional tool to crop it, such as GIMP, Paint.net, or Photoshop.
If the PDF has text and vector content, you can open it in Inkscape, and then delete the content you don't need. Inkscape can export a PNG file, but you can save as a vector file too (SVG for LibreOffice or EMF for Microsoft Word).
If the content you need is a raster image (meaning you can't select the text), you can import the PDF into GIMP and crop it without taking the time to convert it to PNG first.
Use TeX instead
Are you teaching math (since you mentioned the math.se) and getting images of math problems for tests? If yes, I'd strongly recommend learning LaTeX. It's ability to render math is second to none—sometimes even beautiful. There's a tex.se as well. There's even a TeXLive package specifically for creating tests.