Nonlinear Multiple Regressions

I have seven independent variables and would like to test their collective effect on a dependent variable.

When I do a simple least squares regression the results all trend the way I'd expect, but with little statistical significance. I created a scatterplot matrix, and it seems the individual relationships aren't exactly linear. In fact, in each case it appears to be close to the form "Y = root(X)". I tested this, running a nonlinear regression of this form: Y = {b0} + X^{b1}. Sure enough, b1=.48 and is significant.

If I run seven individual regressions, using only one independent variable in each - I get significant results for the exponential betas. My question is now that I can estimate each individual independent variable's nonlinear relationship w/the dependent variable, how can I measure their collective effect? Is this even possible?