I could not resist to write it out:

use strict;
use warnings;
use Math::Factor::XS 'prime_factors';

my $n = shift;
( $n and $n > 0 and $n == int $n ) or die "Need positive integer as in
+put!\n";

my %p;
$p{$_}++ for prime_factors( $n );

my $radical = 1;
$radical *= $_ for grep { $p{$_}%2 } keys %p;
print "sqrt( $n ) = ", sqrt( $n/$radical ), " * sqrt( $radical )\n";
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What I was missing is a product function, like sum from List::Util. There is one in the replacement List::Util that comes with Scalar::Util but that means overwriting the standard module which makes me feel uncomfortable.