If N = \prod_j p_j^{d_j} is the prime factorization of N, its number of divisors \tau(N) is \prod_j (1 + d_j). You want \sqrt{N} to be an integer, so all d_j \ge 2 are even. The minimum ...

sqrt(410)  No simplification exists, Root remains :       • sqrt(410)  Simplify :  sqrt(410)  Factor 410 into its prime factors            410 = 2 • 5 • 41  To simplify a square root, we extract ...

sqrt(416)  Simplified Root :      4 • sqrt(26)  Simplify :  sqrt(416)  Factor 416 into its prime factors            416 = 25 • 13  To simplify a square root, we extract factors which are squares, ...

Alan P. Apr 13, 2015 Both \displaystyle\sqrt{{{4}}} and \displaystyle\sqrt{{{16}}} can be written as rational values \displaystyle\sqrt{{{4}}}={2} and \displaystyle\sqrt{{{16}}}={4} ...

Meave60 Jul 8, 2015 \displaystyle-\sqrt{{4}}=-{2}

One may observe that (\sqrt{2}-1)^2=3-2\sqrt{2} and that (3-2\sqrt{2})^2=17-12\sqrt{2} thus (\sqrt{2}-1)^4=17-12\sqrt{2} giving \sqrt[4]{\alpha}=\sqrt[4]{17-12\sqrt{2}}=\sqrt{2}-1, ...