1. 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.

2. Converting roots into fractional exponents:
Any radical expression may be transformed into an expression with a fractional exponent. The key is to remember that the fractional exponent must be in the form
For example =

3. Negative Exponents:
Remember that a negative in the exponent does not make the number negative!
If a base has a negative exponent, that indicates it is in the “wrong” position in fraction. That base can be moved across the fraction bar and given a postive exponent.
EXAMPLES:

4. Simplifying Radicals by using the Product Rule
                        
This material is based upon work supported by the National Science Foundation under Grant No. DUE
                                                                                                                                                                         STEP- Science, Technology, Engineering, and Mathematics Talent Expansion Program
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Dream Catchers' Math Mentors: Ann Lyndon Kim Ricketts  
                                                               
                     
Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Simplifying Radicals by using the Product Rule
If are real numbers and m is a natural number, then
Examples:
*This one can not be simplified any further due to their indexes (2 and 3) being different!
So, the product of two radicals is the radical of their product!

5. Simplifying Radicals involving Variables:
Let’s get some more practice!

6. Practice:
EX 1:
The index is 2. Square root of 25 is 5. Two goes into 7 three “whole” times, so a
p3 is brought OUTSIDE the radical.The remaining p1 is left underneath the radical.
EX 2:
The index is 4. Four goes into 5 one “whole” time, so a
2 and a are brought OUTSIDE the radical. The remaining 2 and a are left
underneath the radical. Four goes into 7 one “whole” time, so b is brought
outside the radical and the remaining b3 is left underneath the radical.

7. Simplifying Radicals by Using Smaller Indexes:
Sometimes we can rewrite the expression with a rational exponent and “reduce” or simplify using smaller numbers. Then rewrite using radicals with smaller indexes:
More examples:
EX 1:
EX 2:

8. Multiplying Radicals with Difference Indexes:
Sometimes radicals can be MADE to have the same index by rewriting first as rational exponents and getting a common denominator. Then, these rational exponents may be rewritten as radicals with the same index in order to be multiplied.

9. Applications of Radicals:
There are many applications of radicals. However, one of the most widely used applications is the use of the Pythagorean Formula.
You will also be using the Quadratic Formula later in this course!