Name:______________ Block:______________
(Check syllabus & website for quiz and test dates)

In these notes we will learn how to simplify nth roots
So we can solve power equations.
What does it mean to be a "perfect square"?
Write the following as squares (power of 2). Then take the square root. How are roots and radicals related?
Number or variable Written as a square Interpreted as a root
36
16
81 x x x x
8
2
6
1
1
64
27 x x x x x x
___
___
1
x x x x
1
6۰6=(6)
1 x x
___
___
1 x x x x x
1

 x


2
___
___
2
2
Think about the denominator…
 
2۰2۰2=(2) 3 x
2
 
  3 x
 
3
2
36
2
8 x
1 x x
2
2
2

 x
 x

 x
6
1
36
8 x x x



1
2
2
2





2
6
What does it mean to be a "perfect cube"?
Write the following as cubes (power of 3). Then take the cube root. How are roots and radicals related?
Number or variable Written as a square Interpreted as a root
3 1
1
3
3
6
1
1
___
___
___
___
___
___
___
___
3
Think about the denominator…
3 3
3 6
3 1
3
3
6
3

Rational exponents are another way of writing radicals – think POWER over ROOT
POWER (numerator)
Exponential Notation Radical Notation
1 x
Base (x)
Index (Root)
3
ROOT (denominator)
1 x 2
2 x 3
3 x 4 m x or
2 x
1
3
( x )
2 or 3 x
2
4
( x )
3 or 4 3 x n m a n a m 
3 1 x
Exponent (Power) a
 m n
1 a m
, a

0 n
Radicand (Base)
CONVERTING (REWRITING) Rational Exponents to Radical Notation and back!
 How would you write
9
3
2
in radical notation?
 How would you write
125
2
3
in radical notation?
 How would you write
 
5
64
in exponential notation?
 How would you write 3 x
9 in exponential notation?
EVALUATING Rational Exponents and Radical Notation

9
3
2
=

125
2
3
=


5
64
=
3 x
9 =

Evaluate the expression without using the calculator. Use your Exponent Chart
1
144 225 36 2
4 16 3

125
3 2
27
4 3
( 81)

8
1 3
5 243
3 2
9
5 
32
4
2 3
64
16
3
2 32

3
5 256
Writing Radicals in Simplest Form – use your Exponent Chart for perfect nth Roots
45 48
1
4
** You will eventually take a NO calculator section on tests…so start memorizing and recognizing your perfect squares and cubes!!!!
3 48 Find perfect 3 rd Root
Operations with Radicals:
Addition and Subtraction - Need
3

4 3
7 12

5 12
4 48 Find perfect 4 th Root
3 2

8
3 81

3 24

Multiplication - Need

7 6

2
Division - Need
2 12

5 18
20
5
3
3
18
9
5
5
96
3
When we are left with radicals in the denominator… we ____________________________!
With square roots:
5 9 2
2
3
NOW with other roots …The goal is to get a perfect n th root in the denominator to get rid of the radical
3
3
4
We need a perfect cube in the denominator…
3
15
3
4 goes into which perfect cube? _______
So I need to multiply by… _______
We need a perfect cube in the denominator…
3 goes into which perfect cube? _______
So I need to multiply by… _______
3 5
We need a perfect cube in the denominator…
4
2
3 2
2 goes into which perfect cube? _______
So I need to multiply by… _______
8
We need a 4 th root in the denominator…
8 goes into which x 4 from our table? _______
So I need to multiply by… _______

Simplify Expressions with Variables – Factor nth Roots. Leave answers in radical notation.
4 6
16 x y 3
6 9 12
27 x y z 50 x
3
4 5 11
16 x y z 3 27 x y z 12 x z
3 16 w x y 14 4 8 14 5
16 a b c
5 10 7
32 a b

In these notes we will learn how to solve Power & Radical Equations.
What skills do you think this will require?
Solving power equations: A power equation is an equation that involves a variable raised to a power.
Steps to solving a Power Equation
1.
Isolate the exponent first.
2.
Take the appropriate root of both sides. Even Roots:  x Odd Roots: use the sign
3.
Use i when you are taking the SQUARE root of a NEGATIVE #
4.
Check all solutions by plugging your answer into the original equation.
Write an example of a power equation:
When solving, we must follow the reverse order of operations.
1. Add or subtract 2. Multiply or divide 3. Exponents 4. Parentheses
Recall from quadratics…the Square root method:
(x + 3) 2 = 64 3 (x – 5) 2 = - 27
Note: Whenever we solve by square rooting a number, we must put a ______ in front!
This is true for ALL even roots (ie: 4 th root, 6 th root), but NOT true for odd roots. Why is this?
1. Give me a # I can square to get 4.
2. Give me a complex # I can square to get -4.
3. Give me a # I can cube to get 8.
4. Give me a # I can cube to get -8.
5. Give me a # I can raise to the 4 th power to get 81.

2x 4 = 162
6x 3 = 384
1 x
3  
2
4
2x 3 = x 3 + 54
(x – 1) 5 – 3 = –35
(x – 2) 3 = -125
(x – 3) 4 = 625
x 4 = 32
(x + 1) 5 = 100
2(x – 9) 3 = 250

2 x
6 8
3
3 2 x 1 3
4 3 x
  
7 x 15 x 1
Solving radical equations: A radical equation is an equation that involves a variable in the radicand. (Part under the radical symbol).
Steps to solving radical equations:
1.
Isolate the radical on one side of the equation!
2.
Raise each side of the equation to the same power to eliminate the radical. You will be left with a linear, quadratic, or other polynomial equation to solve.
3.
Solve the remaining equation. (use Unit 4 & 5 notes to help you)
4.
Check all solutions by plugging your answer into the original equation.
2 x 4 8 6 x
  
5 x
 
3 x

3
3 5 x
  
2
4 x
 
2
21 x
  
5

In these notes we will learn
 What an inverse function is and how to find it algebraically.
 How to graph Square root and Cube root functions using transformations.
Recall from Unit 2:
An inverse function maps the output values back to their original input values.
For the function
:
{ (2, -2), (3, -3), (5, -5), (7, -7)} a. Create a mapping diagram for the function. b. Create a mapping diagram for the inverse function.
Domain:
Range:
Domain:
Range:
What is the relationship between the domain/range of a function and the domain/range of its inverse?
How could we tell a function and its inverse by the graph?
1.
f(x) = x 2 , x > 0
g(x) = x
2. f(x) = 2x 2 – 3 , x > 0
g(x) =
1
2 x  3

Finding an Inverse Relation from an equation Switch x and y. Solve for y.
Steps:
1.
Switch x and y in equation.
2.
Solve for y.
2. f(x) = -2x + 5 1. y = 4x + 2
3. f(x) = x 2 + 2 , x  0
4. y =
1
2 x 3  2
Verifying that Functions are Inverses of each other:
1. Verify that
Show that f(x) = 4x + 2 and g(x) = f(g(x)) = x
1 x

1 are inverses of each other.
4 2
AND Show that g (f(x)) = x
2. Verify that the inverse functions you found above are correct:

The Square Root Function : The parent function is y
 x
Let’s look at its graph and table of values using our calculator:
 y x y
 x



 x
    

    

What is the domain of the square root parent function?



Why?
What is the inverse of the square root function? Sketch it. Why must we limit the inverse to one branch of the quadratic (parabola)?
The Cube Root Function: The parent function is y

3 x
Let’s look at its graph and table of values using our calculator: x y
 3 x  y



 x
    

    

What is the domain of the square root parent function?
Why?



What is the inverse of the cube root function? Sketch it. Why can we use both braches of the cubic?

Functions have the same transformations as the absolute value function y = a|x – h| + k.
Given y
 a
 f ( x
 h )
 k if a

1
: Vertically Stretch the graph by a factor of a if a

1 : Vertically Shrink the graph by a factor of a if 0 a
 : Reflect the graph about the x-axis
(h, k): Translate the graph horizontally h units and vertically k units.
Let c be a positive real number. Let
Vertical shift c units upward: y

( )
.
( )

( ) c
 Horizontal shift right c units:
Vertical shift c units downward:
( )

(
 c )
( )

( )
 c Horizontal shift left c units: ( )

(
 c )
Transformations of Square Root Function:
 x

2
(h,k): _________





    




 y x
    
 x 1 4
(h,k): _________
 x

3
(h,k): _________





    




 y
  x

2
(h,k): _________ x
    



 y 


 y
    





 x
    

    




 x
    
With SQUARE ROOT FUNCTIONS when you are completing the table of values…you will have xvalues on ONE side of the initial point (h,k). Why?

Transformations of Cube Root Function:
 3 x

4
(h,k): _________





    




 y x
    

3 x 2 3
(h,k): _________





    




 y x
     y

3 3 x
(h,k): _________




 y x
         






3 x

2
(h,k): _________
    









 y x
    

In these notes we will ANALYZE the graphs of Square Root and Cube Root Functions
1. Dividing by zero is undefined: a denominator can NEVER be equal to zero.
2. The square root of a negative number does not exist . . . we NEVER put a negative number under a square root (unless we are dealing in complex numbers).
We will look at Case #1 in Unit 7.
Case #2 above: No Negatives Under the Radical Sign!! x  0
Do you have a square root? Do you have a rational power that has a denominator of 2?
If not, then you don’t have to worry about this restriction.
1 f(x) = x f(x) = x 2 Domain: The set of all real numbers x ≥ 0 y
 x

5 y
 x

3 ( )
 y
 x

4
Now let’s go back and define our characteristics from Unit 2 with the square root and cube root function.
The Square Root Function : The parent function is
2 x

3 y
 x
  1
2
(h, k): _____ x - Intercepts: ____________ y - intercept: ______________



 y
Domain: _____________________
 x
Range: _______________________
Increasing: ____________________
Decreasing: ___________________
    





    

The Cube Root Function: The parent function is y

3 x
(h, k): _____ x - Intercepts: ____________ y - intercept: ______________




 y
Domain: _____________________ x
Range: _______________________
    


    
Increasing: ____________________ 

Decreasing: ___________________ 
Complete the following. Graph without a calculator. Then verify with your calculator and use to find your intercepts if necessary. Round to the nearest tenth.
 x

4
y
 
2 x

3
(h,k): __________




 y x
(h,k): __________




 y x x-int: ___________ y-int: ___________
Domain: _________
Range: __________
    





     x-int: ___________ y-int: ___________
Domain: _________
Range: __________
    





     y
 x 2 5
(h,k): __________ x-int: ___________ y-int: ___________
Domain: _________
Range: __________





    




 y x
     y

3 x 4 1
(h,k): __________ x-int: ___________ y-int: ___________
Domain: _________
Range: __________





    




 y x
    