Here's an Algorithm::Loops solution, then, based on lidden's nifty solution:

use strict; 
use warnings;
use List::Util qw[ reduce ];
use Algorithm::Loops 'NextPermuteNum';

sub closest{
    my $exact = sqrt(shift);
    my $best = 1;
    do {
        my $guess = 1;
        for (@_) {
        $guess *= $_;
            $best = $guess if (abs($exact - $guess) < abs($exact - $be
+st));
            last if $guess > $exact;
        }
    } while NextPermuteNum(@_);
    $best;
}

my @pfs = (2,2,3,3,5,5,7);
# Compute product of factors
our ($a, $b);
my $num = reduce { $a * $b } @pfs;
my $root = sqrt $num;
print "N=$num, R=$root\n";

print closest($num, @pfs), "\n";
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