UNIT #6: RADICAL FUNCTIONS 7-1: ROOTS AND RADICAL EXPRESSIONS Essential Question: When is it necessary to use absolute value signs in simplifying radicals?

7-1: Roots and Radical Expressions  Definitions  Since 5 2 = 25, we say that 5 is a square root of 25  Since 5 3 = 125, we say that 5 is a cube root of 125  Since 5 4 = 625, we say that 5 is a fourth root of 625  Since 5 5 = 3125, we say that 5 is a fifth root of 3125

7-1: Roots and Radical Expressions  Real numbers with even roots can have 0, 1, or 2 solutions (just like the discriminant)  The 4 th root of 16 can be 2 or -2, since (2) 4 = (-2) 4 = 16  The 6 th root of -16 does not exist, as there is no number x such that x 6 = -16  The n th root of 0 is always 0.  Real numbers with odd roots can only have one solution  The cube root of -125 is -5, since (-5) 3 = -125  (5) 3 = 125, so there is no duplication with odd powers.  A chart summarizing the rules of roots is on the next slide

7-1: Roots and Radical Expressions Type of NumberNumber of Real n th Roots When n is Even Number of Real n th Roots When n is Odd positive negativenone1 How to calculate n th roots on your calculator: - Your calculator should have a button that looks like this: - First enter what root power you’re looking for, then the button, then the number you’re trying to find. - Example: Find all real cube roots of Enter: Your calculator will only give you the positive root for even roots, you will have to remember about the negative option (+)

7-1: Roots and Radical Expressions  Find the cube root(s) of  Find the cube root(s) of 1 / 27  Find the fourth root(s) of 1  Find the fourth root(s) of  Find the fourth root(s) of 16 / 81

7-1: Roots and Radical Expressions  A weird quirk about roots  Notice that if x = 5,  And when x = -5,  There needs to be some way to handle this situation  So if, at any time: Both the root and exponent underneath a radical are even And the output exponent is odd  The variable must be protected inside absolute value signs

7-1: Roots and Radical Expressions  Examples using (or not using) absolute values  The square (2) root of a 6 th power comes out to be an odd power, absolute value signs must be used  Finding the cube (3) root of a problem means absolute values signs aren’t necessary at any point  Finding the 4 th root means absolute value signs may be necessary. The x comes out to the 1 st (odd) power, so it gets absolute value signs, while the y (even power) does not.

7-1: Roots and Radical Expressions  Your turn: 

7-1: Roots and Radical Expressions  Assignment  Page 372, 1-28 (all problems)  Due Tomorrow