For questions involving radical of numbers or radical of expressions (i.e. numbers/expressions raised to the power of a fraction).
A radical expression is any mathematical expression containing a radical symbol $~(√~)~$.
Many people mistakenly call this a 'square root' symbol, and many times it is used to determine the square root of a number. However, it can also be used to describe a cube root, a fourth root, or higher.
When the radical symbol is used to denote any root other than a square root, there will be a superscript number in the $'V'$-shaped part of the symbol. For example, $~3\sqrt{8}~$ means to find the cube root of $~8~$. If there is no superscript number, the radical expression is calling for the square root.
The term underneath the radical symbol is called the radicand.
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. Click on the link to see some examples of Prime Factorization. Also, factor any variables inside the radical.
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 not 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.
A closely related tag is the nested-radicals tag.
