How to Simplify Radicals

To simplify radicals, rather than looking for perfect squares or perfect cubes within a number or a variable the way it is shown in most books, I choose to do the problems a different way, and here is how.
Here are the steps required for Simplifying Radicals:



Step 1:
Find the prime factorization of the number inside the radical. Start by dividing the number by the first prime number 2 and continue dividing by 2 until you get a decimal or remainder. Then divide by 3, 5, 7, etc. until the only numbers left are prime numbers. 

Step 2:
Determine the index of the radical. The index tells you how many of a kind you need to put together to be able to move that number or variable from inside the radical to outside the radical. For example, if the index is 2 (a square root), then you need two of a kind to move from inside the radical to outside the radical. If the index is 3 (a cube root), then you need three of a kind to move from inside the radical to outside the radical.
Step 3:
Move each group of numbers or variables from inside the radical to outside the radical. If there are nor enough numbers or variables to make a group of two, three, or whatever is needed, then leave those numbers or variables inside the radical. Notice that each group of numbers or variables gets written once when they move outside the radical because they are now one group.
Step 4:
Simplify the expressions both inside and outside the radical by multiplying. Multiply all numbers and variables inside the radical together. Multiply all numbers and variables outside the radical together.


Example 1 – Simplify:

Step 1: Find the prime factorization of the number inside the radical.

Step 2: Determine the index of the radical. In this case, the index is two because it is a square root, which means we need two of a kind.

Step 3: Move each group of numbers or variables from inside the radical to outside the radical. In this case, the pair of 2’s and 3’s moved outside the radical.

Step 4: Simplify the expressions both inside and outside the radical by multiplying.

Example 2 – Simplify: 

Step 1: Find the prime factorization of the number inside the radical and factor each variable inside the radical.



Step 2: Determine the index of the radical. In this case, the index is three because it is a cube root, which means we need three of a kind.   



Step 3: Move each group of numbers or variables from inside the radical to outside the radical. In this case, the 3’s, x’s (two groups), and y’s moved outside the radical.   


 
Step 4: Simplify the expressions both inside and outside the radical by multiplying.


Example 3 – Simplify:

Step 1: Find the prime factorization of the number inside the radical and factor each variable inside the radical.   

 
Step 2: Determine the index of the radical. In this case, the index is five because it is a fifth root, which means we need five of a kind.     



Step 3: Move each group of numbers or variables from inside the radical to outside the radical. In this case, the 3’s, x’s, and y’s moved outside the radical.     




Step 4: Simplify the expressions both inside and outside the radical by multiplying.


Example 4 – Simplify:

Step 1: Find the prime factorization of the number inside the radical and factor each variable inside the radical.     


Step 2: Determine the index of the radical. In this case, the index is two because it is a square root, which means we need two of a kind.     



Step 3: Move each group of numbers or variables from inside the radical to outside the radical. In this case, the 2’s, 3’s, x’s (two groups), and y’s (four groups) moved outside the radical.     



Step 4: Simplify the expressions both inside and outside the radical by multiplying.    


Example 5 – Simplify:

Step 1: Find the prime factorization of the number inside the radical and factor each variable inside the radical.     


Step 2: Determine the index of the radical. In this case, the index is four because it is a fourth root, which means we need four of a kind.     



Step 3: Move each group of numbers or variables from inside the radical to outside the radical. In this case, the pair of 2’s and y ’s moved outside the radical.   


 
Step 4: Simplify the expressions both inside and outside the radical by multiplying.    



Example 6 – Simplify:

Step 1: Find the prime factorization of the number inside the radical and factor each variable inside the radical.     


Step 2: Determine the index of the radical. In this case, the index is three because it is a cube root, which means we need three of a kind.     



Step 3: Move each group of numbers or variables from inside the radical to outside the radical. In this case, the 2’s, 3’s, x’s, and y’s (two groups) moved outside the radical.     



Step 4: Simplify the expressions both inside and outside the radical by multiplying.    


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