91Ó°ÊÓ

Prime factorization is the process of breaking down a number into its prime components. Primes are numbers greater than 1 that have no divisors other than 1 and themselves. To simplify a radical, identifying these prime factors is essential because it helps us find the perfect squares within the number.
To factorize 90, we start by dividing it by the smallest prime, which is 2. This gives us 45, which is not divisible by 2, so we move to the next prime, 3. Dividing 45 by 3 gives us 15, and divides again by 3 to result in 5, which is a prime.
Thus, the prime factorization of 90 is written as:

• 90 = 2 \( \times \) 3\(^2\) \( \times \) 5

Finding prime factors helps simplify radicals by identifying perfect squares, leading to easier computation of square roots.

A perfect square is a number that can be expressed as the product of an integer with itself. Recognizing perfect squares is crucial when simplifying radicals because we need to take their square roots.
In the case of the number 90, its prime factorization is 2 \( \times \) 3\(^2\) \( \times \) 5. The term \( 3^2 \) is a perfect square, as 3 \( \times \) 3 equals 9. This component can be swapped out with its root, 3.
This means:

• \( \sqrt{3^2} = 3 \)

By identifying perfect squares, we simplify the expression under the radical, making calculations simpler and more straightforward.

The square root of a number is a value that, when multiplied by itself, gives the original number. Understanding this concept is key when you are asked to simplify radicals.
The main objective in simplifying radicals is reducing the expression to its lowest terms. With the previous example of simplifying \( \sqrt{90} \), we utilized prime factorization and recognized perfect squares to achieve this:

• The prime factorization of 90 highlighted the perfect square, \( 3^2 \).
• We extracted \( 3 \) from the radical because \( \sqrt{3^2} = 3 \).
• The expression became \( 3\sqrt{10} \) as we took 3 out of the square root.

Always remember that the square root's goal is to find a number that can simplify the radical as much as possible, helping you solve radicals with ease.