Download 6-1 Evaluate nth Roots and Use Rational Exponents

6-1 Evaluate nth Roots and Use Rational Exponents
Name_____________________
Objective: To evaluate nth roots and use rational exponents.
Algebra 2 Standard 12.0
*Real nth Roots of a.
: If x n  a, then x is nth root of a .
when n is an even integer
x n  a, then x   n a
when n is an odd integer
x n  a, then x  n a
a<0
No real nth roots.
Ex) x 2  5 , then ___________.
One negative real nth root: : x = n a  a1/n
Ex) x3  8, then ___________.
a=0
One real nth root: x = n 0  0
Ex) x 2  0, then ___________.
One real nth root: : x = n 0  0
Ex) x3  0, then ___________.
a>0
Two real nth roots: : x =  n a  a1/n
Ex) x 2  5, then ___________.
One positive real nth root: : x = n a  a1/n
Ex) x3  8, then ___________.
Example 1: Find the nth root(s) of a. (use x n  a form)
a) n = 5, a = -32
b) n = 6, a = 1
You Try: Find the indicated real nth root(s) of a.
a) n = 6, a = 64
b) n = 3, a = -64
*Rational Exponents: Let a1/n be an nth root of a, and let m be a positive integer.
Recall:
na
m
 1
a   an  
 
m
n
1
 an
 a
n
m
a m/ n 
1
a
m/ n
Example 2: Evaluate:
a) 1252/3
b) 84/3
You Try: Evaluate.
a) 45/2
b) 9 1/ 2

1
a 
1/ n
m

1
 a
n
m
(a  0)
Example 3: Evaluate the expression using a calculator. Round the result to two decimal places when
appropriate.
a) 221/4
Alg 2 Adv_Ch.6 Notes-page 1
b) 355/6
c)
 11 
5
4
You Try: Evaluate the expression using a calculator. Round the result to two decimal places
when appropriate.
b) 652/3
a) 42/5
Example 4: Solve each equation.
a) 6 x 3  384
b)
c)
 x  8
5

3
30

2
 100
Example 5: An exercise ball is made from 7854 square centimeters of material. Find the diameter of the
ball. (Use the formula S  4 r 2 for the surface area of the sphere.)
You Try: Solve.
1
a) x 5  512
2
c)
 x  2
3
 14
Alg 2 Adv_Ch.6 Notes-page 2
b) 3x 2  108
d)
 x  5
4
 16
6-2 Apply Properties of Rational Exponents
Name________________
Objective: To simplify expressions involving rational exponents.
Algebra 2 Standard 12.0
*Properties of Rational Exponents (same as those from 5-1):
Let a and b be real numbers and let m and n be rational numbers.
Property
Example
a m  a n  a m n
When multiplying powers
with the same base, add the exponents.
a 
m
 ab 
n
 a mn
When raising a power to a power,
multiply the exponents.
m
 a mb m
To find a power of a product,
apply the exponent for each factor.
1
,a  0
am
To change negative exponent to positive,
write the reciprocal of the base.
am 
am
 a mn , a  0
an
m
am
a
   m ,b  0
b
b
When dividing powers
with the same base, subtract the exponents.
To find a power of a quotient,
apply the exponent for the numerator and denominator.
Example 1: Use the properties of rational exponents to simplify the expression.
5
 71/4
a. 121/8 125/6
b.
10
d.
102/5
 561/4 
e.  1/4 
 7 
1/3

3
c.
2
6
 46

5
You Try: Simplify.
3/4
a. 2
2
1/2
Alg 2 Adv_Ch.6 Notes-page 3
3
b. 1/ 4
3
 201/2 
c.  1/2 
 5 
3
1/6
Example 2: The ratio of the magnitudes of two earthquakes with magnitude m1 and m2 (as given be the
10m1
Richter scale) is given by the equation r  m2 . The table gives the magnitudes of the some of the
10
largest earthquakes that have occurred in the U.S. How many times stronger was the 1964 quake in
Alaska than the 1812 quake in Missouri?
Year
1812
1906
1958
1964
State
MO
CA
AK
AK
Magnitude
7.9
7.7
8.3
9.2
*Properties of Radicals:
Product Property of Radicals:
Quotient Property of Radicals:
n
n
a b  n a  n b
a na

,b  0
b nb
Example 3: Use the properties of radicals to simplify the expression.
a.
3
5
250  3 16
b.
96
5
3
You Try: Use the properties of radicals to simplify the expression.
a.
3
4
12  18
3
b.
4
80
5
*A radical with an index n is in simplest form
if 1) the radicand has no perfect nth powers as factors
2) there is no radical in the denominator
3) there is no negative exponents
Example 4: Write the expression in simplest form.
4
10
a. 3 104
b. 4
27
You Try: Simplify the expression.
a.
4
27  4 3
Alg 2 Adv_Ch.6 Notes-page 4
b.
5
3
4
3
c.
250
3
2
* Radical expressions with the _____________ index and radicand are like radicals. To add or
subtract like radicals, use the distributive property.
Example 5: Simplify the expression.
a. 7 5 12  5 12
b. 4  92/3   8  92/3 
c.
3
81  3 24
You Try: Simplify the expression.
a.
3
5  3 40
* Because a variable can be positive, negative, or zero, sometimes absolute value is needed
when simplifying a variable expression. If it is assumed that all variables are positive, you do
not need to worry about absolute value.
Example 6: Simplify the expression. Assume all variables are positive.
a.
4
625z12
b.
 32m n 
c.
6
d.
56ab3/4
7a5/6c 3
5 30
1/5
r6
s18
You Try: Write the expression in simplest form. Assume all variables are positive.
a.
3
b.
7
6x 4 y 9 z14
p8
q5
Alg 2 Adv_Ch.6 Notes-page 5
Example 8: Perform the indicated operation. Assume all variables are positive.
a. 18 3 u  113 u
b. 15a 4b 2/3  8a 4b 2/3
c. 10 4 5x7  x 4 80 x3
You Try: Simplify the expression. Assume all variables are positive.
27q 9
a.
3
b.
5
c.
6 xy 3/4
3x1/2 y1/2
d.
x10
y5
9w5  w w3
Alg 2 Adv_Ch.6 Notes-page 6
6-3 Perform Function Operations and Composition
Name__________________
Objective: To perform operations with functions.
Algebra 2 Standards 24.0 and 25.0
*Operations on Functions: Let f and g be any two functions. A new function h can be defined by
performing any of the four basic operations on f and g.
Operation
Addition
Definition
h(x) = f(x) + g (x)
Subtraction
Multiplication
Division
h(x) = f(x) – g (x)
h(x) = f(x) · g (x)
h(x) =
Example: f(x) = 5x, g(x) = x + 2
f ( x)
g ( x)
The domain of h consists of the x-values that are in the domains of both f and g.
The domain of the quotient does not include x-values for which g(x) = 0.
*Power Function: y  axb where a is a real number and b is a rational number.
Example 1: Let f ( x)  5 x1/3 and g ( x)  11x1/3 . Find the following.
a. f(x) + g(x)
b. f(x) – g(x)
c. the domains of parts a and b
You Try: Let f ( x)  2 x 2/3 and g ( x)  7 x 2/3 . Find the following.
a. f(x) + g(x)
b. f(x) – g(x)
c. the domains of parts a and b
Example 2: Let f( x) = 8x and g(x) = 2x5/6. Find the following and state the domain.
f ( x)
a. f(x) · g (x)
b.
g ( x)
You Try: Let f( x) = 3x and g(x) = x1/5. Find the following and state the domain.
f ( x)
a. f(x) · g (x)
b.
g ( x)
Alg 2 Adv_Ch.6 Notes-page 7
Example 3: A small company sells computer printers over the Internet. The company’s total monthly
revenue (R) and costs (C) are modeled by the functions R(x) = 120x and C(x) = 2500 + 75x where x is the
number of printers sold.
a. Find R(x) – C(x)
b. Explain what this difference means.
*Composition of Functions:
The composition of a function g
with a function f is:
Domain of f
h(x) = g(f(x))
Range of f
The domain of h is the set of all
x-values such that x is in the domain
of f and f(x) is in the domain of g.
Domain of g
Example 4: Let f(x) = 3x – 4 and g(x) = x2 – 1. What is the value of f(g(-3))?
A.
B.
C.
D.
-34
8
20
168
You Try: Let f(x) = 3x – 8 and g(x) = 2x2. Find the following.
a. g(f(5))
b. f(g(5))
c. f(f(5))
d. g(g(5))
Alg 2 Adv_Ch.6 Notes-page 8
Range of g .
Example 5: Your starting wage for your part-time job was $6 an hour. All employees get a 5% raise after
6 months. You are given an additional raise of $.75 per hour as a reward for your outstanding work. Find
your new hourly wage if the 5% raise is applied before the $.75 raise.
You Try: Use example 5 and find your new hourly wage if the $.75 raise is applied before the 5% raise.
Example 6: Let f(x) = 6x 2 and g(x) = 4x + 5. Find the following.
a. f(g(x))
b. g(f(x))
c. g(g(x))
You Try: Let f(x) = 2x 1 and g(x) = 2x + 7. Find f(g(x)), g(f(x)) and f(f(x)).
Alg 2 Adv_Ch.6 Notes-page 9
6-4 Use Inverse Functions
Name:__________________
Objective: To find inverse functions.
Algebra 2 Standards 24.0 and 25.0
*Recall that an inverse relation just switches the x and y values (i.e., your domain becomes your range
and your range becomes the domain). The graph of an inverse relation is a
______________________________ of the original graph.
*To find an inverse relation, switch x and y and then solve for y.
Example 1: Find the equation for the inverse of the relation y = 4x + 2.
*Inverse Functions: Functions f and g are inverses of each other provided:
f(g(x)) = x
and
g(f(x)) = x
1
The function g is denoted by f and read “f inverse.”
Example 2: Verify that f(x) = 4x + 2 and f 1 ( x) 
1
1
x
4
2
Example 3: Find the inverse of f(x) = 2x – 1 Then verify that your result and the original function are
inverses.
You Try: Find the inverse of the given function. Then verify that your result and the original function are
inverses.
a. f(x) = x + 4
Alg 2 Adv_Ch.6 Notes-page 10
b. f(x) = -3x + 1
*When finding inverses of models (real life examples), do not switch the variables because the
letters represent the specific quantities. Just solve for the other variable.
Example 4: A small company produces greeting cards. The cost C (in dollars) of producing n greeting
cards per month can be modeled by the function C = 360 + 0.60n.
a. Find the inverse of the model.
b. Use the inverse function to find the number of greeting cards produced in a month in which the
company’s total cost to produce the cards was $615.
*Inverses of Nonlinear Functions: Copy the original graph and find the reflection in the line y = x.
Notice that the inverse of g(x) = x3 is a function, but that the inverse of f(x) = x2 is not a function.
f(x) = x2
g(x) = x3
*If the domain of f(x) = x2 is restricted to only nonnegative real numbers,
then the inverse is a function.
Example 5: Find the inverse of f(x) = x2 + 2, x < 0. Then graph f and f 1 .
Alg 2 Adv_Ch.6 Notes-page 11
*You can use the graph of a function f to determine whether the inverse of f is a function
by applying the ________________________ line test. The inverse of a function f is also a function
if and only if no horizontal line intersects the graph of f more than once.
Example 6: Consider the function f(x) = 3x5 – 2. Determine whether the inverse of f is a function. Then
find the inverse.
You Try: Find the inverse of the function. Then graph the function and its inverse.
a. f(x) =  x 3  4
b. f(x) = 2x5 + 3
Example 7: The population of a town can be modeled by P  16,500t 0.15 , where t is the number of years since
1998. Find the inverse model that gives the number of years as a function on the population.
Alg 2 Adv_Ch.6 Notes-page 12
6-5 Graph Square Root and Cube root Functions
Name:__________________
Objective: To find inverse functions.
Algebra 2 Standards 24.0 and 25.0
1) Square Root function
f ( x)  x
 Domain is x  0
 Range is y  0
2) Cube root Function
g ( x)  3 x
Domain and Range are
all real numbers.
3) Transformation of Radical Functions
f ( x)  x
y a xh k


a  changes steepness :
f ( x)  3 x

y  a3 x  h  k
a  1 stretches f(x) vertically.
0  a  1 shrinks f(x) vertically.
negative a reflects f(x) over the x-axis.


h  shifts left or right:
k  shifts up or down
Example 1: Graph the function and state the domain and range.
a. y  2 x
Alg 2 Adv_Ch.6 Notes-page 13
b. y  
13
x
2
You Try: Graph the function and state the domain and range.
a. y  2 3 x
b. y  2 x
Example 2: Graph y  3 x  3  2 and state the domain and range.
Example 3: Graph y  2 3 x  3  2 and state the domain and range.
Alg 2 Adv_Ch.6 Notes-page 14
6-6 Solve Radical Equations
Name: _______________________
Objective: To solve radical equations.
Algebra 2 Standard 12.1
*Equations that have variables inside radicals are called radical equations.
*How to solve 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.
3. Solve the equation.
4. Check your solution.
Ex. 1: Solve
a. 5 x  9  11
b.
You Try: Solve
a. x  25  4
b. 2 3 x  3  4
3
x  9  1
Ex. 2: If an 8-foot ladder is leaned against a wall, the height is given by h  x   64  x2 ,
where x is the distance along the floor from the base of the ladder to the wall. Estimate the distance that
the ladder should be placed from the wall in order for the ladder to reach 7 feet off the floor.
Alg 2 Adv_Ch.6 Notes-page 15
Ex. 4: Solve  x  1
Ex. 3: Solve 7 x3/5  56
You Try:
Solve  x  4 
2/3
 9  16
You Try:
 x  2
1/3
2/3
 6  10
3 7
*Raising each side of an equation to the same power sometimes creates extraneous solutions.
This is a solution that appears to be a solution, but when substituted back into the original
equation, does not actually work. Be sure to check all apparent solutions when you have
variables more than one places.
Ex. 5: Solve
x 6  x 8
You Try: Solve
2x  5  x  7
Alg 2 Adv_Ch.6 Notes-page 16
Ex. 6. Solve
x9  x  3
You Try: Solve
x6 2  x2