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:

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. 