We investigate dimensionality reduction for Euclidean \(k\)-means and \(k\)-medians clustering. We show that the cost of the optimal solution for \(k\)-means and \(k\)-medians is preserved up to a factor of \( (1+\varepsilon) \) under a projection onto a random \( \log k\)dimensional subspace. Further, the cost of every clustering is preserved within \( (1+\varepsilon) \) Our bound on the dimension is nearly optimal. Additionally, our result applies to Euclidean \(k\)-clustering with the distances raised to the \(p\)-th power for any constant \(p\). Joint work with Yury Makarychev and Ilya Razenshteyn.