Simplifcation of $p$-norm to the $ p$

Question

If define the p-norm as $$||x||_p = (\sum_{i=1}^{N} |x_i|_p)^\frac{1}{p}$$ Can I state that $$||x||_p^p = ((\sum_{i=1}^{N} |x_i|_p)^\frac{1}{p})p = (\sum_{i=1}^{N} |x_i|_p)$$ This feels like a gross oversimplification, but I cannot seem to get it out of my head.