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Unit #6: Radical Functions 7-1: Roots and Radical Expressions

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

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Unit #6: Radical Functions 7-1: Roots and Radical Expressions

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  1. Unit #6: Radical Functions7-1: Roots and Radical Expressions Essential Question: When is it necessary to use absolute value signs in simplifying radicals?

  2. 7-1: Roots and Radical Expressions • Definitions • Since 52 = 25, we say that 5 is a square root of 25 • Since 53 = 125, we say that 5 is a cube root of 125 • Since 54 = 625, we say that 5 is a fourth root of 625 • Since 55 = 3125, we say that 5 is a fifth root of 3125

  3. 7-1: Roots and Radical Expressions • Real numbers with even roots can have 0, 1, or 2 solutions (just like the discriminant) • The 4th root of 16 can be 2 or -2, since (2)4 = (-2)4 = 16 • The 6th root of -16 does not exist, as there is no number x such that x6 = -16 • The nth 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

  4. 7-1: Roots and Radical Expressions How to calculate nth 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 0.008 - Enter: Your calculator will only give you the positive root for even roots, you will have to remember about the negative option (+)

  5. 7-1: Roots and Radical Expressions • Find the cube root(s) of -1000 • Find the cube root(s) of 1/27 • Find the fourth root(s) of 1 • Find the fourth root(s) of -0.0001 • Find the fourth root(s) of 16/81

  6. 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. 7-1: Roots and Radical Expressions • Examples using (or not using) absolute values • The square (2) root of a 6th 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 4th root means absolute value signs may be necessary. The x comes out to the 1st (odd) power, so it gets absolute value signs, while the y (even power) does not.

  8. 7-1: Roots and Radical Expressions • Your turn:

  9. 7-1: Roots and Radical Expressions • Assignment • Page 372, 1-28 (all problems) • Due Tomorrow

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