Simplifying Radicals

Perfect Squares

= 2 = 4 = 5 = 10 = 12

= = = = = = = = = = Perfect Square Factor * Other Factor LEAVE IN RADICAL FORM

= = = = = = = = = = Perfect Square Factor * Other Factor LEAVE IN RADICAL FORM

+ To combine radicals: combine the coefficients of like radicals

Simplify each expression

Simplify each expression: Simplify each radical first and then combine.

= = = = = = = = = = Perfect Square Factor * Other Factor LEAVE IN RADICAL FORM

Simplify each expression

* To multiply radicals: multiply the coefficients and then multiply the radicands and then simplify the remaining radicals.

Multiply and then simplify

= l X l = l Y 3 l = P 2 X 3 Y = 2X 2 Y = 5C 4 D 5 But…Let’s assume all of the variables are positive : Note: When we have an even index !!!

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.

= = = =

A.11h B.11h 2 C.13h 2 D.–11h

Over Lesson 6–4 5-Minute Check 2 A. B.–4ay 3 C. D.8ay 3

5-Minute Check 5 A.about 1.43 m B.about 2.52 m C.about 3.11 m D.about 5.48 m

Simplify Expressions with the Product Property Factor into squares where possible. Product Property of Radicals Answer:Simplify.