I'm trying to simplify this equation...: $$ t~=~\frac{\left( \sum_{i=1}^{n} x_iy_i \right) \left( \sqrt{ n - 1} \right) } {\left( \sqrt{ \left( \sum_{i'=1}^{n} x^2_{i'} \right) \left( \sum_{i=1}^{n} y^2_i \right) - \sum_{i=1}^{n} \left( 2x_iy_i\hat{\beta} + x_i^2\hat{\beta}^2 \right)} \right)} $$ ...to this: $$ t~=~\frac{\left( \sum_{i=1}^{n} x_iy_i \right) \left( \sqrt{ n - 1} \right) } {\sqrt{ \left( \sum_{i'=1}^{n} x^2_{i'} \right) \left( \sum_{i=1}^{n} y^2_i \right) - \left( \sum_{i=1}^{n} x_iy_i \right)^2}} $$ which, as I see it, comes down to simplifying...: $$ \sum_{i=1}^{n} \left( 2x_iy_i\hat{\beta} + x_i^2\hat{\beta}^2 \right) $$ ...to: $$ \left( \sum_{i=1}^{n} x_iy_i \right)^2 $$
I've looked here for multiple summation identities, however the one I can't find (and feel as though is key to answering this problem) is the simplification of: $$ \sum_{i=1}^{n} x_iy_i $$ Is there a simplification? Something along the lines of $ \sum_{i=1}^{n}x_i\sum_{i=1}^{n}y_i $?
Note: Problem Context - I'm attempting to show that the t-statistic for coefficient $\hat{\beta}$ when Y is regressed onto X without an intercept can be re-written as the second equation. The information given in the problem is: $$ \hat{\beta}~=~\frac{\sum_{i=1}^{n} x_iy_i}{\sum_{i'=1}^n x^2_{i'}}~~~~~and~~~~~SE(\beta)~=~\sqrt{\frac{ \sum_{i=1}^{n} \left( y_i - x_i\hat{\beta} \right)^2 }{\left( n - 1 \right) \sum_{i'=1}^{n} x^2_{i'}}}~~~~~and~~~~~t~=~\frac{\hat{\beta}} {SE\left( \hat{\beta} \right)} $$