How is the Solow residual measured?
Below please find a portion of a lecture slide a professor of mine used last year. Please note that $\gamma_{\tilde{y}}$ denotes per-capita output growth, $\gamma_{\tilde{k}}$ denotes per-capita capital growth and $\alpha(t)$ denotes the income share of capital. $R(t)$ - the Solow residual - can then be easily computed.

The production function you have stated was the one we used for this example. So it can be applied here directly.



Source: "Growth and Business Cycles" lecture by Professor Gerhard Sorger, University of Vienna