Let \(k\gt 0\) be an integer and \(t\ge (\sqrt{2e})^k\). Then \(t\gt 0\) and \(t^{2/k}/e\ge 2\).

Let \(k\gt 0\) be an integer and \(t\gt 0\), and put \(q=t^{2/k}/e\). Then \(((\sqrt q)^k/t)^q=\exp (-kt^{2/k}/(2e))\).

For an integer \(k\gt 0\) and reals \(\lambda ,t\gt 0\), \((t(\sqrt\lambda )^k)^{2/k}=\lambda t^{2/k}\).

Let \(k\gt 0\) be an integer, \(\lambda \gt 0\), and \(t\ge (\sqrt{2e/\lambda })^k\). Then \(t\gt 0\) and \(\lambda t^{2/k}/e\ge 2\).

Let \(k\gt 0\) be an integer and \(\lambda ,t\gt 0\), and put \(q=\lambda t^{2/k}/e\). Then \(((\sqrt q\lambda ^{1/q-1/2})^k/t)^q=\lambda ^k\exp (-k\lambda t^{2/k}/(2e))\).