[Tex/LaTex] runaway argument with fraction

math-mode

I am trying to write the following equation:

\begin{equation}
    \text{$N_R$} \text{x} (\text{$\Delta\sigma_R$})^m = 2 \text{x} {10^6} \text{x} (\text{$\Delta\sigma_C$})^m    with  m = 3 for \text{N} \leq 5 \text{x} {10^6}
    \text{$N_R$} \text{x} (\text{$\Delta\tau_R$})^m = 2 \text{x} {10^6} \text{x} (\text{$\Delta\tau_C$})^m    with  m = 5 for \text{N} \leq 5 \text{x} {10^6}
    where \text{$\Delta\sigma_D$} = \text{$\frac{2}{5}^{\frac{1}{3}$} \text{x} \text{$\Delta\sigma_C$} = 0.7368 \text{x} \text{$\Delta\sigma_C$} is the constant amplitude fatigue limit
    \end{equation}

But getting error as:

Runaway argument?
{$\frac{2}{5}^{\frac{1}{3}$} \text{x} \text{$\Delta\sigma_C$} =\ETC.
! Paragraph ended before \text@ was complete.
<to be read again>
             \par

May please help….

Best Answer

You might be looking for the following, but i am really not sure.

\documentclass{article}
\usepackage{mathtools}
\begin{document}
\begin{align}
    N_R \times (\Delta\sigma_R)^m &= 2 \times {10^6} \times (\Delta\sigma_C)^m    
&&\text{with~}  m = 3 \text{~for~} N\leq 5 \times {10^6}\\
N_R \times (\Delta\tau_R)^m &= 2 \times {10^6} \times (\Delta\tau_C)^m    
&&\text{with~}  m = 5 \text{~for~} N \leq 5 \times {10^6}
\end{align}
where $\Delta\sigma_D = \left(\frac{2}{5}\right)^{\frac{1}{3}} \times \Delta\sigma_C = 0.7368 \times \Delta\sigma_C $
is the constant amplitude fatigue limit
\end{document}
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