1. Radical Expressions
Finding a root of a number is the inverse operation of raising a number to a power.
radical sign
index
radicand
This symbol is the radical or the radical sign
The expression under the radical sign is the radicand.
The index defines the root to be taken.

2. Rewriting Radicals

3. EX:

4. Simplifying Radical Expressions
For a radical expression to be simplified it has to satisfy the following conditions:
The radicand has no factor raised to a power greater than or equal to the index. (EX:There are no perfect-square factors.)
The radicand has no fractions.
No denominator contains a radical.
Exponents in the radicand and the index of the radical have no common factor, other than one.

5. Steps
1. Find the largest perfect square( cube, etc.) which will divide evenly into the number under your radical sign. This means that when you divide, you get no remainders, no decimals, no fractions
2. Write the number appearing under your radical as the product (multiplication) of the perfect square and your answer from dividing.
3. Give each number in the product its own radical sign
4. Reduce the "perfect" radical which you have now created

8. Operations with Radicals

9. Adding and Subtracting
You must have like radicals before adding and subtracting radical expressions
Like radicals are radicals with the same index and the same radicand.
If the radicals are the same you can simply add/subtract the number in front of the radical and keep the radical the same. If the radicals are not the same try and simplify to where the radicals are the same and then add/sub.

10. Ex:

12. Multiplying and Dividing
When multiplying radicals, you must multiply the numbers OUTSIDE the radicals AND then multiply the numbers INSIDE the radicals
When dividing radicals, you must divide the numbers OUTSIDE the radicals AND then divide the numbers INSIDE the radicals.
You do not have to have like radicals as you do when you add and subtract

13. Ex:

14. Rationalizing the denominator
Should a radical appear in the denominator of a fraction, it will need to be "removed" if you are trying to simplify the expression.
To "remove" a radical from the denominator, multiply the top and bottom of the fraction by that same radical to create a rational number (a perfect square radical) in the denominator.
This process is called rationalizing the denominator.

15. Ex:

16. Conjugate