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Prime factorization is the process of breaking down a number into its simplest building blocks, which are prime numbers. Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. To find the prime factorization of a number like 72, you start by dividing it by the smallest prime number (2), and continue dividing by prime numbers until you reach 1. For example:

• 72 ÷ 2 = 36
• 36 ÷ 2 = 18
• 18 ÷ 2 = 9
• 9 ÷ 3 = 3
• 3 ÷ 3 = 1

From this process, we see that the prime factors of 72 are 2, 2, 2, 3, and 3. We can write this as: \[ 72 = 2^3 \times 3^2 \]This breakdown is essential before you can simplify a radical.

A perfect square is a number that can be expressed as the square of an integer. For example, 4, 9, and 16 are perfect squares because they are the squares of 2, 3, and 4 respectively (\[ 2^2 = 4, \ 3^2 = 9, \ 4^2 = 16\]). In the context of simplifying radicals, identifying perfect squares within the radicand (the number inside the radical) allows you to simplify the expression more easily. In our solution for simplifying \( \sqrt{72} \), we identified \( 3^2 \) as a perfect square within the prime factorization of 72. This step helps in breaking down the radical into simpler parts:\[ \sqrt{72} = \sqrt{2^3 \times 3^2} = \sqrt{2^2 \times 2 \times 3^2} = \sqrt{2^2} \times \sqrt{3^2} \times \sqrt{2} \]This shows that recognizing perfect squares makes simplifying radicals much easier.

The radicand is the number inside the radical symbol \( \sqrt{} \). In the example \( \sqrt{72} \), 72 is the radicand. To simplify the radical, you need to focus on factorizing the radicand into its prime numbers. After the prime factorization, you can then identify perfect square factors.
A clear understanding of the radicand is crucial because it guides the steps to simplifying the radical. By factorizing 72, we saw that its prime factorization is \( 2^3 \times 3^2 \). Breaking it down into perfect squares and non-perfect squares gave us a simpler radical expression: \( 6 \sqrt{2} \).
Keep in mind, the process is straightforward:

• Identify the radicand.
• Perform prime factorization.
• Break down into perfect squares and simplify.

With these steps, you can simplify any radical expression with ease.