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How can I remove space automatically added above expression under square root when I write for example\sqrt{k^2_t-k^2}. In fraction\frac{\sqrt{k^2_t-k^2}}{1} there isn't this type of space.

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    Welcome to TeX.SE. It depends on where you are writing the first square root. if you type $\sqrt{k^2_t-k^2}$ you'll get the root with no additional space, since expressions between $...$ (or \(...\)) are in-line mode. Otherwise if you type \[\sqrt{k^2_t-k^2}\] you'll get the root with space, because it is displaystyle mode. – Cragfelt Jan 14 '18 at 23:39
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    You can remove it by using \textstyle prior to \sqrt, note however that this affacts more than just that space. – Skillmon Jan 14 '18 at 23:53
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When TeX is in display-math mode, the numerator and denominator of a fraction are rendered in text style. If you want to apply text style to other material in a display-math environment, you need to provide the directive \textstyle explicitly.

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\documentclass{article} 
\begin{document}
\[ % start an unnumbered display math group
\sqrt{k^2_t-k^2} \qquad
{\textstyle\sqrt{k^2_t-k^2}} \qquad
\frac{\sqrt{k^2_t-k^2}}{1}
\]
\end{document}

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