UNIT #6: RADICAL FUNCTIONS7-1: 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 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

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

7-1: Roots and Radical Expressions

Type of Number

Number of Real nth Roots When

n is Even

Number of Real nth Roots When

n is Odd

positive 2 1

0 1 1

negative none 1How 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 (+)

x

3 0.008 0.23 0.008x

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

10

13

1 1or

no real fourth root

2 23 3or

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

2 2(5) 25 5x x

2 2( 5) 25 5x x

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.

6 34 2x x

3 3 6 2a b ab

4 8 24 x y x y

7-1: Roots and Radical Expressions Your turn:

2 44x y 22 x y

3 627c 23c

8 124 x y 2 3x y

7-1: Roots and Radical Expressions Assignment

Page 372, 1-28 (all problems) Due Tomorrow