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Prime factorization is the process of expressing a number as a product of its prime numbers. Prime numbers are those that can only be divided by 1 and themselves without leaving a remainder. When simplifying radicals, prime factorization helps in identifying which numbers can be "paired" to move outside the radical sign.

For example, let's consider the number 72. Its prime factors are found by dividing it by the smallest prime numbers. Start by dividing by 2 until it's no longer possible:

• 72 divided by 2 is 36
• 36 divided by 2 is 18
• 18 divided by 2 is 9

Now that 9 can't be divided by 2 anymore, switch to the next smallest prime, which is 3:

• 9 divided by 3 is 3, and finally
• 3 divided by 3 is 1

So, the prime factorization of 72 is written as \(2^3 \times 3^2\). This means that when we use 72 inside a square root, we can express it as \(\sqrt{2^3 \times 3^2}\), allowing for further simplification.

Understanding square roots is crucial to simplifying expressions involving them. A square root \(\sqrt{x}\) represents a number which, when squared (multiplied by itself), returns \(x\). The goal when simplifying square roots is often to "pull out" perfect squares.

When simplifying \(\sqrt{72xy}\), we first identify the perfect squares within the factorization. From the prime factorization \(\sqrt{2^3 \times 3^2 \times x \times y}\):

• The pair \(2^2\) can be pulled out as 2.
• Similarly, \(3^2\) can come out as 3.

So, \(\sqrt{72xy}\) simplifies to \(6\sqrt{2xy}\), where we multiply the numbers outside the square root (2 and 3), leaving any remaining factors (\(\sqrt{2xy}\)) inside the square root.

When variables are involved, only pairs of variables can be taken out of the square root as well. If no pairs exist, they stay inside, just like the numbers.

Dividing expressions that contain square roots can seem tricky, but it follows basic division principles. Particularly, you can divide both numbers outside the radicals and those inside if they share the same index.

In this problem, we start with dividing \(\frac{6\sqrt{2xy}}{2\sqrt{2}}\). Division involves these steps:

• Divide the coefficients (the numbers outside the square roots): here, 6 divided by 2 equals 3.
• Cancel out the same terms within the square root if possible: \(\sqrt{2}\) appears both in numerator and denominator, so they cancel out.

Thus, simplifying \(\frac{6\sqrt{2xy}}{2\sqrt{2}}\) results in \(3\sqrt{xy}\). This simplification works due to the basic mathematical principle that any number divided by itself equals one, which simplifies terms inside the radicals.