Numerically speaking, the square root of a number (n) is the quantity that produces n when multiplied by itself. Geometrically speaking, the square root of a number (n) is the length of a square whose area is equal to n.

Finding the prime factorization of n allows you to see the all of the primes (the atomic parts, if you will) inside the number. These primes can be meted out to two identical piles. Any primes that are left over will have to be given out to the two piles in the form of their irrational square roots. For example, an extra 2 left over will have to be fairly given out as a √2 for each pile.

A full-blown example problem would be the task of finding the square root of 180. The prime factorization of 180 = 2²∙3²∙5¹. The even number of 2s and 3s can easily be split up into two piles, but there's only one 5, which will have to be split up into a pair of √5s.

180 = 2²∙3²∙(√5)² = (2∙3∙√5)(2∙3∙√5) = (6√5)(6√5).

√180 = √(36∙5) = √36∙√5 = 6√5.

In other words, the sides of a square with area of 180 square units must be 6√5, which is approximately 6(2.236...) = 13.4 units long.

Hope this helps!

Thank you, Pei.