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nth Roots and Rational Exponents
What you should learn:
Evaluate nth roots of real numbers
using both radical notation and
rational exponent notation
Evaluate the expression.
Solving Equations.
6.1 nth Roots and Rational Exponents
Using Rational Exponent Notation
Rewrite the expression using RATIONAL EXPONENT notation.
 If n is odd, then a has one real nth root:
Ex)
3
125
125 3
1
a a
n
1
n
5
simplify
If n is even and a > 0, then a has two real nth roots:
Ex)
4
81
 81
1
4
3
 a  a
 If n is even and a = 0, then a has one nth root:
n
n
1
n
0 0 n 0
1
 If n is even and a < 0, then a has NO Real roots:
6.1 nth Roots and Rational Exponents
Using Rational Exponent Notation
Rewrite the expression using RADICAL notation.
Ex)
Ex)
24 3
1
28
1
4
3
4
24
28
6.1 nth Roots and Rational Exponents
Evaluating Expressions
Evaluate the expression.
Ex)
Ex)
9
3
32
2
2
 9
3
1
5
32
2

5
 3
 27
1
1
 2
2
3
 32 
5
2
1

4
6.1 nth Roots and Rational Exponents
Solving Equations
Ex)
4
4
x  81
4
x   81
4
2 x  64
5
Ex)
5
5
x 5  32
x  32
5
x  3
When the exponent is EVEN you
must use the Plus/Minus
x2
When the exponent is ODD you
don’t use the Plus/Minus
6.1 nth Roots and Rational Exponents
Solving Equations
Ex)
4
4
( x  4)  256
4
Take the Root 1st.
x  4   256
4
x  4  4
x4  4
x 8
Very Important
2 answers !
x  4  4
x0
6.1 nth Roots and Rational Exponents
Evaluate the expressions.
 625
4
4
 625
6.1 nth Roots and Rational Exponents
Properties of Rational Exponents
6.2 Properties of Rational Exponents
Review of Properties of Exponents from section 5.1
Page 420 in your book
am * an = am+n
(am)n = amn
(ab)m = ambm
1
-m
a =a
a
m-n
=
a
a
a = a
 
m
m
n
m
b
m
bm
Ex: Simplify. (no decimal answers)
c. (43  23 )
a.
61/2 * 61/3
1
= 61/2 + 1/3
= 63/6 + 2/6
= 65/6
(2 2 )3
(2 )
2 3
2
(2  2 )
6
b. (271/3 * 61/4)2
= (271/3)2 * (61/4)2
= (3)2 * 62/4
= 9 * 61/2
3
d.
3
 14
 18
 1
 94

1





3
3

1
3
 (29 ) 13  23  13  1
8
2
3

18
9
3
4
3
4
3
4
3
 18 
    24
9
** All of these examples were in rational exponent form to begin with, so the
answers should be in the same form!
Ex: Simplify.
3
25

5=
a.
3
3
Ex: Write the expression in
simplest form.
25 5
3
a.
= 125 = 5
4
4
64 =
16  4=
4
= 24 4
3
32
3
4
b.
=
3
=
3
32
4
b.
4
8= 2
=
4
=
** If the problem is
in radical form to
begin with, the
answer should be in
radical form as well.
7
8
4
4
4
7
8
7 42
4 =
8
2
2.828
Can’t have a Radical
in the basement!
4
14
4
16
4
=
14

2
16  4 4
.967
Ex: Perform the indicated operation
a. 5(43/4) – 3(43/4)
c. 3 625  3 5
= 2(43/4)
= 3 125  5  3 5
3
3
= 5 5 5
3
b. 3 81  3 3
=6 5
= 3 27  3  3 3
= 33 3  3 3
If the original problem is in radical form,
3
=2 3
the answer should be in radical form as well.
If the problem is in rational exponent form, the
answer should be in rational exponent form.
Simplify the Expressions
Ex)
x 2 3  x3 5
Ex)
x 
Ex)
x2 3
14
x
Ex)
13
3
x
2
5
5
x
2 3 3 5
 x19 15
x 2 15
x
2 31 4
x
53
x
x
33
5 12
x
23
 x  x2 3
 x3 x 2
6.2 Rational Exponents and Radical Functions
More Examples
a. x 2  x
b.
c.
d.
6
x6 
x
11
y11 
y
4
r  (r )
8
8
1
4
r
8
4
 r2
Directions: Simplify the Expression.
Assume all variables are positive.
a. 3
27z 9  3 27  3 z 9
 3z
3
d.
c.
5
x

y2
5
5
1
4 3
6r t
b. (16g4h2)1/2
= 161/2g4/2h2/2
= 4g2h
x5

10
y
18rs
2
3
3r
3
4
2
3 3
3r s t
x5
y10
1
1
4
2
3
s t3
Yes, Change both to rational exponent form and use the
quotient property.
6.2 Properties of Rational Exponents
A2.2.5
6.3 Power Functions and Functions Operations
Operations on Functions: for any two functions f(x) & g(x)
1. Addition
h(x) = f(x) + g(x)
2. Subtraction h(x) = f(x) – g(x)
3. Multiplication h(x) = f(x)g(x)
4. Division
h(x) = f(x)/g(x)
5. Composition h(x) = f(g(x))
OR
g(f(x))
** Domain – all real x-values that “make sense” (i.e. can’t
have a zero in the denominator, can’t take the even nth
root of a negative number, etc.)
Example: Let f(x) = 3x1/3 & g(x) = 2x1/3
Find (a) the sum, (b) the difference, and (c) the domain for each.
(a)
3x1/3
+
2x1/3
= 5x1/3
(b) 3x1/3 – 2x1/3
= x1/3
(c) Domain of (a) all real
numbers
Domain of (b) all real
numbers
The SUM
f ( x)  g ( x)
The DIFFERENCE
f ( x)  g ( x)
Ex: Let f(x) = 4x1/3 & g(x) = x1/2.
Find (a) the product, (b) the quotient, and (c) the
domain for each.
a.)
4x  x
1
3
4x
b.)
5
1
The PRODUCT
2
f ( x)  g ( x)
6
The QUOTIENT
1
4x 3
1
4
6
1

4
x
 1
x 2
6
x
4
6
x
f ( x)
g ( x)
Ex: Let f(x) = 4x1/3 & g(x) = x1/2.
Find (a) the product, (b) the quotient, and (c) the domain for
each.
(a)
(b)
4x1/3
4x
x
*
x1/2
1
3
=
4x5/6
4
 x
5
Domain of (a) all reals ≥ 0, because
you can’t take the 6th root of a
negative number.
1
2
= 4x1/3-1/2
= 4x-1/6
= 4
x
=
4x1/3+1/2
6
Domain of (b) all reals > 0, because
you can’t take the 6th root of a
negative number and you can’t have a
4 denominator of zero.
1
6
6
x
Cont’
6.3 Perform Function Operations and Composition
Composition
• f(g(x)) means you take the function g and
plug it in for the x-values in the function f, then
simplify.
• g(f(x)) means you take the function f and
plug it in for the x-values in the function g,
then simplify.
You purchase a baseball glove with a price tag of
$180 dollars. The sports store applies a newspaper
coupon of $50 and a 10% store discount.
A
When the
coupon is
applied before
the discount.
We will visit this
question later…
B
When the
discount is
applied before
the coupon.
Let f(x) = 2x-1 & g(x) = x2 - 1.
Ex:
Find (a) f(g(x)), (b) g(f(x)), (c) f(f(x)), and (d) the domain of each.
f(g(x))
(a) 2(x2-1)-1 =
2
x2 1
f(f(x)) (c) 2(2x-1)-1
= 2(2-1x)
= 2x  x
2
g(f(x))
(b) (2x-1)2-1
=
22x-2-1
(d) Domain of (a) all reals except x = ±1.
Domain of (b) all reals except x = 0.
=
4
1
2
x
Domain of (c) all reals except x = 0, because
2x-1 can’t have x = 0.
You purchase a baseball glove with a price tag of
$180 dollars. The sports store applies a newspaper
coupon of $50 and a 10% store discount.
A
Find the final
price of the
purchase when
the coupon is
applied before
the discount.
B
Find the final
price when the
discount is
applied before
the coupon.
STEP 1
Write functions for the discounts.
Function for $50 coupon: f(x) = x - 50
Function for 10% discount: g(x) = x – 0.10x = 0.90x
STEP 2
Compose the functions
$50 coupon is applied first:
g(f(x)) = g(x – 50) = 0.90(x - 50)
10% discount is applied first:
f(g(x)) = f(0.90x) = 0.90x - 50
STEP 3
Evaluate the functions g(f(x)) f(g(x)) when x = 180
$50 coupon is applied first:
g(f(180)) = 0.90(180 – 50) = $117
10% discount is applied first:
f(g(180)) = 0.90(180) – 50 = $112
The final price is $117 when the $50 coupon is
applied before the 10% discount.
The final price is $112 when the 10% discount is
applied before the $50 coupon.
f ( x )  3 x  2 g ( x)   x
2
x2
h( x ) 
5
#21) g ( f ( 2))
f ( x)  3 x
1
#29) g ( f ( x))
g ( x)  2 x  7 h( x )  x  4
3
How is the composition of functions different form
the product of functions?
The composition of functions is a function of a function. The output of one function
becomes the input of the other function. The product of functions is the product of the
output of each function when you multiply the two functions.
6.3 Power Functions and Functions Operations
Inverse Functions
What you should learn:
Find inverses of linear functions.
Verify that f and g are inverse functions.
Graph the function f. Then use the
graph to determine whether the inverse
of f is a function.
Michigan Standard A2.2.6
6.4 Inverse Functions
Temperature Conversion
The formula to convert
temperatures from degrees
Celsius-to-Fahrenheit is…
9
F  C  32
5
But, how do you convert from
Fahrenheit-to-Celsius??
C=
Toon in later…
Review from chapter 2
• Relation – a mapping of input values (x-values)
onto output values (y-values).
• Here are 3 ways to show the same relation.
y = x2
Equation
Table of
values
Graph
x
y
-2
4
-1
1
0
0
1
1
• Inverse relation – just think: switch the x & y-values.
x = y2
x
y
y x
4
-2
1
-1
0
0
1
1
* the inverse of
an equation:
switch the x & y
and solve for y.
** the inverse of
a table:
switch the x & y.
** the inverse of a graph:
the reflection of the original
graph in the line y = x.
To find the inverse of a function:
1. Change the f(x) to a y.
2. Switch the x & y values.
3. Solve the new equation for y.
** Remember functions have to pass the
vertical line test!
Ex: Find an inverse of f(x) = -3x+6.
3 Steps: - change f(x) with y
-switch x & y
-solve for y
y = -3x + 6
x = -3y + 6
x - 6 = -3y
x6
y
3
1
y  x2
3
Verify Inverse Functions
• Given 2 functions, f(x) & g(x) ,
• if f(g(x)) = x AND g(f(x)) = x,
• then f(x) & g(x) are inverses of each other.
Symbols: f -1(x) means “f inverse of x”
Ex: Verify that f(x)= -4x+8 and g(x) = -1/4x + 2
are inverses.
• Meaning find f(g(x)) and g(f(x)). If they both equal x,
then they are inverses.
f(g(x))= -4(-1/4x + 2) + 8
g(f(x))= -1/4(-4x + 8) + 2
= x–8+8
=x–2+2
=x
=x
** Because f(g(x))= x and g(f(x)) = x, they are inverses.
Temperature Conversion
The formula to convert
temperatures from degrees
Celsius-to-Fahrenheit is…
9
F  C  32
5
But, how do you convert from
Fahrenheit-to-Celsius??
C=
Temperature Conversion
9
F  C  32
Solve for C
5
- 32
- 32
5 F  32  9 C 5


5
9
9
5
( F  32)  C
9
Ex:
(a)
Find the inverse of f(x) = x5.
(b) Is f -1(x) a function?
1. y = x5
2. x = y5
3. 5 x  5 y 5
5
xy
y x
5
Yes , f -1(x) is a
function.
(hint: look at the graph!
Does it pass the vertical line
test?)
Horizontal Line Test
• Used to determine whether a function’s
inverse will be a function by seeing if the
original function passes the horizontal
line test.
• If the original function passes the
horizontal line test, then its inverse is a
function.
• If the original function does not pass the
horizontal line test, then its inverse is not
a function.
Ex: Graph the function f(x)=x2 and
determine whether its inverse is a function.
Graph does not pass the
horizontal line test,
therefore the inverse is not
a function.
Ex: f(x)=2x2 - 4 Determine whether f -1(x) is a
function, then find the inverse equation.
y = 2x2 - 4
x = 2y2 - 4
x + 4 = 2y2
x4
 y2
2
x4
y
2
f -1(x) is not a function.
OR, if you fix the
tent in the
basement…
1
y  x2
2
Ex: g(x)=2x3
y = 2x3
x = 2y3
x
 y3
2
3
x
 y
2
y
Inverse is a function!
3
OR, if you fix the
tent in the
basement…
x
2
y
3
4x
2
Finding Domain and Range
State the domain and range of the function.
1)
y x
x0
y0
2)
yx
2
x
All real #’s
y0
Solving Radical Equations
What you should learn:
Solve equations that contain
Radicals.
Solve equations that contain
Rational exponents.
Michigan Standard L1.2.1
6.6 Solving Radical Equations
Solve equations that contain Radicals
Solve the equation. Check for extraneous solutions.
Simple Radical
check your
solutions!!
Ex.1)
Key Step:
x 3
 x   3
2
2
To raise each side of the
equation to the same
power.
x 9
6.6 Solving Radical Equations
Simple Radical
Ex.2)
3
x  6  12
6 6
3
Key Step:
x 6
 x   6
3
3
x
Before raising each side to the
same power, you should isolate
the radical expression on one
side of the equation.
3
 216
6.6 Solving Radical Equations
One Radical
Ex.3)
2(46)  8  4  6
2x  8  4  6
4 4

2 x  8 2 10

100  4  6
2 x  8  10 
2 x  8  100
8
2
10  4  6
8
2x  92
2
2
x  46
6.6 Solving Radical Equations
Two Radicals
Ex.4)
12  2(2)  2 2  0
12  2 x  2 x  0
2 x 2 x
 12  2 x   2 x 
12  2 x 2 2 x
12  2x  4x
 2x
8 2 2  0
2
2 2 2 2  0
 2x
12  6x
6
6
2 x
6.6 Solving Radical Equations
Radicals with an Extraneous Solution
What is an Extraneous Solution?
… is a solution to an equation raised to a
power that is not a solution to the original
equation.
Example 5)
x  3  4x
6.6 Solving Radical Equations
Radicals with an Extraneous Solution
Ex.5)
x  3  4x
6.6 Solving Radical Equations
Radicals with an Extraneous Solution
Ex.5)
x  3  4x
x  3
2

 4x 
9  3  4(9)
2
x  6 x  9  4x
2
 4x
 4x
x  10 x  9  0
2
( x  9)( x  1)  0
x 9
1  3  4(1)
x 1
6.6 Solving Radical Equations
Reflection on the Section
Without solving, explain why
2 x  4  8 has no solution.
6.6 Solving Radical Equations
Solve equations that contain Rational exponents.
Ex. 6)
x
x 
52
52 25
 32
(4)
52
 32
 32 
x 4
25
32  32
5 2
 ?
2 5
it
6.6 Solving Radical Equations
Solve equations that contain Rational exponents.
Ex. 7)
2x
32
2
32
32 23
x 
x
 250
2
 125 2 3
 125

x  52

13 2
x  125
it
2(25)
32
 250
2(253 )1 2
12
2(15625)
2(125)
 250
x  25
6.6 Solving Radical Equations
Reflection on the Section
Without solving, explain why
2 x  4  8 has no solution.
6.6 Solving Radical Equations