Simplifying Radical Expressions For a radical expression to be simplified it has to satisfy the following conditions: 1. The radicand has no factor raised to a power greater than or equal to the index. (EX:There are no perfect-square factors.) 2. The radicand has no fractions. 3. No denominator contains a radical. 4. Exponents in the radicand and the index of the radical have no common factor, other than one.

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 =

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:

Simplifying Radicals by using the Product Rule If are real numbers and m is a natural number, then 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 NOTE: Some of the forms on this page may require the Adobe Acrobat Reader software. Most may already have this plug- in. If you do not, you can download it for free by clicking on the Adobe icon to the right. Dream Catchers' Math Mentors:Ann LyndonKim Ricketts Dream Catchers' Math Mentors:Ann LyndonKim 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.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. So, the product of two radicals is the radical of their product! Examples: *This one can not be simplified any further due to their indexes (2 and 3) being different!

Simplifying Radicals involving Variables: Examples: This is really what is taking place, however, we usually don’t show all of these steps! The easiest thing to do is to divide the exponents of the radicand by the index. Any “whole parts” come outside the radical. “Remainder parts” stay underneath the radical. For instance, 3 goes into 7 two whole times.. Thus will be brought outside the radical. There would be one factor of y remaining that stays under the radical. Let’s get some more practice!

Practice: The index is 2. Square root of 25 is 5. Two goes into 7 three “whole” times, so a p 3 is brought OUTSIDE the radical.The remaining p 1 is left underneath the radical. EX 1: 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 b 3 is left underneath the radical.

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:

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.

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! Both of these formulas have radicals in them. To learn more about them you may go to: Pythagorean Theorem What is the Pythagorean Formula? Quadratic FormulaPythagorean Theorem What is the Pythagorean Formula? Quadratic Formula