PLEASE COMPLETE THE
PREREQUISITE SKILLS
PG 412 #1-12
CHAPTER 6:
RATIONAL EXPONENTS AND
RADICAL FUNCTIONS
BIG IDEAS:
 USE RATIONAL EXPONENTS
 PERFORMING FUNCTION
OPERATIONS AND FINDING INVERSE
FUNCTIONS
• SOLVING RADICAL EQUATIONS
LESSON 1:
EVALUATE NTH ROOTS AND
USE RATIONAL EXPONENTS
WHAT IS THE RELATIONSHIP
BETWEEN NTH ROOTS AND
RATIONAL EXPONENTS?
ESSENTIAL QUESTION
VOCABULARY
• Nth root of a: For an integer n greater
than 1, if bn = a, then b is an nth root of
a. written as 𝑛 𝑎
• Index of a radical: The integer n, greater
than 1, in the expression 𝑛 𝑎
EXAMPLE 1
Find nth roots
Find the indicated real nth root(s) of a.
a.
n = 3, a = –216
b.
n = 4, a = 81
SOLUTION
a. Because n = 3 is odd and a = –216 < 0, –216 has
one real
cube root. Because (–6)3 = –216, you
can write = 3√ –216 = –6 or
(–216)1/3 = –6.
b. Because n = 4 is even and a = 81 > 0, 81 has two real fourth
roots. Because 34 = 81 and
(–3)4 = 81, you can write ±4√ 81
= ±3
Evaluate expressions with rational exponents
EXAMPLE 2
Evaluate (a) 163/2 and (b) 32–3/5.
SOLUTION
Rational Exponent Form
= (161/2)3 = 43 = 64
a.
163/2
b.
32–3/5 =
=
1
=
323/5
1
23
=
1
(321/5)3
1
8
Radical Form
3
163/2 = ( 16 )
32–3/5
=
=
1
323/5
1
=
23
=
= 43 = 64
1
(5 32 )3
1
8
EXAMPLE 3
Approximate roots with a calculator
Keystrokes
Expression
a.
91/5
b.
c.
(
Display
9
1
5
1.551845574
123/8
12
3
8
2.539176951
4
7
3
4
4.303517071
 7)3 = 73/4
for Examples 1, 2 and 3
GUIDED PRACTICE
Find the indicated real nth root(s) of a.
1.
n = 4, a = 625
SOLUTION
2.
3.
±5
n = 6, a = 64
SOLUTION
SOLUTION
4.
±2
n = 3, a = –64.
–4
n = 5, a = 243
SOLUTION
3
for Examples 1, 2 and 3
GUIDED PRACTICE
Evaluate expressions without using a calculator.
5. 45/2
SOLUTION
6.
7. 813/4
32
9–1/2
SOLUTION
SOLUTION
27
8. 17/8
1
3
SOLUTION
1
GUIDED PRACTICE
for Examples 1, 2 and 3
Evaluate the expression using a calculator. Round the result to two
decimal places when appropriate.
Expression
9.
42/5
SOLUTION
1.74
10.
–
64 2/3
SOLUTION
0.06
11.
(4√ 16)5
SOLUTION
32
12.
(3√ –30)2
SOLUTION
9.65
EXAMPLE 4
Solve equations using nth roots
Solve the equation.
a.
4x5
= 128
x5 = 32
Divide each side by 4.
5 32
x = 
Take fifth root of each side.
x = 2
Simplify.
Solve equations using nth roots
EXAMPLE 4
b.
(x – 3)4
= 21
4
x–3 = +
–  21
Take fourth roots of
each side.
4
x = +
– 21 + 3
x =
4
x
5.14
21 + 3
or
Add 3 to each side.
or
x
x = –421 + 3
0.86
Write solutions
separately.
Use a calculator.
WHAT IS THE RELATIONSHIP
BETWEEN NTH ROOTS AND
RATIONAL EXPONENTS?
The nth root of a can be written as a to the
1
𝑎
ESSENTIAL QUESTION
SIMPLIFY THE EXPRESSION:
43*48
LESSON 2:
APPLY PROPERTIES OF
RATIONAL EXPONENTS
HOW ARE THE PROPERTIES
OF RATIONAL EXPONENTS
RELATED TO PROPERTIES
OF INTEGER EXPONENTS?
ESSENTIAL QUESTION
VOCABULARY
• Simplest form of a radical: A radical with
index n is in simplest form if the radicand
has no perfect nth powers as factors
and any denominator has been
rationalized
• Like radicals: Radical expressions with
the same index and radicand
Use properties of exponents
EXAMPLE 1
Use the properties of rational exponents to simplify the expression.
a.
71/4 71/2
b.
(61/2 41/3)2
c.
(45 35)–1/5
d.
e.
5
51/3
=
= (61/2)2 (41/3)2
= [(4 3)5]–1/5
51
51/3
421/3 2
61/3
= 73/4
= 7(1/4 + 1/2)
=
= 5(1 – 1/3)
42
6
1/3
= 6(1/2 2) 4(1/3 2)
= (125)–1/5
= 12[5
(–1/5)]
= 52/3
2
= (71/3)2
= 7(1/3 2)
= 72/3
= 61 42/3
= 6 42/3
= 12 –1 = 1
12
Use properties of radicals
EXAMPLE 3
Use the properties of radicals to simplify the expression.
a.
b.
12
3
4
4
80
5
3
18 =
=
4
3
12 18
80
5
=
4
=
16
3
=
216 = 6
2
Product property
Quotient property
Write radicals in simplest form
EXAMPLE 4
Write the expression in simplest form.
a.
3
 135
3
=
 27
5
=
 27
3
3
=
3
3 5
 5
Factor out perfect cube.
Product property
Simplify.
Write radicals in simplest form
EXAMPLE 4
5
b.
 7
5
 8
=
5
 7
5
5
5
 8
 4
 4
Make denominator a perfect fifth
power.
5
=
 28
Product property
5
 32
5
=
 28
2
Simplify.
Add and subtract like radicals and roots
EXAMPLE 5
Simplify the expression.
4
4
4
4
a.
 10 + 7  10 = (1 + 7)  10 = 8  10
b.
2 (81/5) + 10 (81/5) = (2 +10)
c.
3
3
3
 54 –  2 =  27
(81/5) = 12 (81/5)
3
3
3
3
3
 2 –  2 = 3  2 –  2 = (3 – 1)  2 = 2  2
3
for Examples 3, 4, and 5
GUIDED PRACTICE
Simplify the expression.
6.
4
27
4
3
8.
5
3
4
5
SOLUTION
7.
3
 24
SOLUTION
2
3
 250
9.
3
 5
3
+  40
3
 2
SOLUTION
5
SOLUTION
3
3 5
EXAMPLE 6
Simplify expressions involving variables
Simplify the expression. Assume all variables are positive.
3
a.
3
b.
(27p3q12)1/3 = 271/3(p3)1/3(q12)1/3
c.
d.
3
 64y6 =  43(y2)3
4
m4
n8
4
=
14xy 1/3
2x 3/4 z –6

4

m4
n8
4
=
3
=  43
 m4
4

(n2)4
=
= 7x(1 – 3/4)y1/3z –(–6)
 (y2)3
= 3p(3
= 4y2
1/3)q(12
m
n2
= 7x1/4y1/3z6
1/3)
= 3pq4
EXAMPLE 7
Write variable expressions in simplest form
Write the expression in simplest form. Assume all variables are positive.
a.
5
 4a8b14c5
5
=  4a5a3b10b4c5
5
=  a5b10c5
b.
3
x
y8
=
=
3
5
=
ab2c  4a3b4
x
y
y8 y
3
5
 4a3b4
Product property
Simplify.
Make denominator a perfect cube.
xy
y9
Factor out perfect fifth
powers.
Simplify.
EXAMPLE 7
Write variable expressions in simplest form
3
=
 xy
3
 y9
Quotient property
3
=
xy
y3
Simplify.
EXAMPLE 8
Add and subtract expressions involving variables
Perform the indicated operation. Assume all variables are positive.
a.
b.
c.
1
3
w +
w =
5
5
3xy1/4 – 8xy1/4
3
1
+
5
5
= (3 – 8) xy1/4
w
=
4
w
5
= –5xy1/4
3
3
3
3
2
z
5
–


54z
= 12z  2z2 – 3z  2z2
12 2z
3
= (12z – 3z)  2z2
3
= 9z  2z2
for Examples 6, 7, and 8
GUIDED PRACTICE
Simplify the expression. Assume all variables are positive.
10.
3
 27q9
SOLUTION
11.
5
12.
3q3
x10
3x 1/2 y 1/2
SOLUTION
13.
y5
SOLUTION
6xy 3/4
x2
y
2x1/2y1/4
 9w5 – w  w3
SOLUTION
2w2  w
HOW ARE THE PROPERTIES
OF RATIONAL EXPONENTS
RELATED TO PROPERTIES
OF INTEGER EXPONENTS?
All properties of integer exponents also apply to
rational exponents
ESSENTIAL QUESTION
LET F(X) = 3X + 5.
FIND F(-6)
LESSON 3
PERFORM FUNCTION
OPERATIONS AND
COMPOSITION
WHAT OPERATIONS CAN
BE PERFORMED ON A PAIR
OF FUNCTIONS TO OBTAIN
A THIRD FUNCTION?
ESSENTIAL QUESTION
VOCABULARY
• Power Function: A function of the form
y=axb, where a is a real number and b is
a rational number
• Composition: The composition of a
function g with a function f is h(x) =
f(f(x)).
EXAMPLE 1
Add and subtract functions
Let f (x) = 4x1/2 and g(x) = –9x1/2. Find the following.
a.
f(x) + g(x)
SOLUTION
f (x) + g(x)
b.
= 4x1/2 + (–9x1/2)
= [4 + (–9)]x1/2
= –5x1/2
f(x) – g(x)
SOLUTION
f (x) – g(x)
= 4x1/2 – (–9x1/2)
= [4 – (–9)]x1/2
= 13x1/2
EXAMPLE 1
c.
Add and subtract functions
the domains of f + g and f – g
SOLUTION
The functions f and g each have the same domain: all
nonnegative real numbers. So, the domains of f + g and f – g also
consist of all nonnegative real numbers.
Multiply and divide functions
EXAMPLE 2
Let f (x) = 6x and g(x) = x3/4. Find the following.
a.
f (x)
g(x)
SOLUTION
f (x)
b.
g(x)
= (6x)(x3/4)
= 6x(1 + 3/4) = 6x7/4
f (x)
g(x)
SOLUTION
f (x)
g(x)
=
6x
x3/4
= 6x(1 – 3/4) = 6x1/4
EXAMPLE 2
c.
Multiply and divide functions
the domains of f
g and
f
g
SOLUTION
The domain of f consists of all real numbers, and the domain of g
consists of all nonnegative real numbers. So, the domain of f g
consists of all nonnegative real numbers. Because g(0) = 0, the
domain of
is restricted to all positive real numbers.
f
g
EXAMPLE 3
Solve a multi-step problem
Rhinos
For a white rhino, heart rate r (in beats per minute) and life span s (in
minutes) are related to body mass m (in kilograms) by these functions:
r(m) = 241m–0.25
•
•
s(m) = (6
Find r(m) s(m).
Explain what this product represents.
106)m0.2
EXAMPLE 3
Solve a multi-step problem
SOLUTION
STEP 1
Find and simplify r(m) s(m).
r(m) s(m)
= 241m –0.25 [ (6
106)m0.2 ]
= 241(6 106)m(–0.25 + 0.2)
= (1446
106)m
= (1.446
109)m –0.05
–0.05
Write product of r(m)
and s(m).
Product of powers
property
Simplify.
Use scientific notation.
EXAMPLE 3
Solve a multi-step problem
STEP 2
Interpret r(m)
s(m).
Multiplying heart rate by life span gives the total number of heartbeats for
a white rhino over its entire lifetime.
GUIDED PRACTICE
for Examples 1, 2, and 3
Let f (x) = –2x2/3 and g(x) = 7x2/3. Find the following.
1.
f (x) + g(x)
SOLUTION
f (x) + g(x)
2.
= –2x2/3 + 7x2/3
= (–2 + 7)x2/3
= –2x2/3 – 7x2/3
= [–2 + ( –7)]x2/3
= 5x2/3
f (x) – g(x)
SOLUTION
f (x) – g(x)
= –9x2/3
GUIDED PRACTICE
3.
for Examples 1, 2, and 3
the domains of f + g and f – g
SOLUTION
all real numbers; all real numbers
for Examples 1, 2, and 3
GUIDED PRACTICE
Let f (x) = 3x and g(x) = x1/5. Find the following.
4.
f (x)
SOLUTION
5.
g(x)
3x6/5
f (x)
g(x)
SOLUTION
3x4/5
GUIDED PRACTICE
6.
the domains of f
for Examples 1, 2, and 3
g and
f
g
SOLUTION
all real numbers; all real numbers except x=0.
GUIDED PRACTICE
for Examples 1, 2, and 3
Rhinos
7.
Use the result of Example 3 to find a white rhino’s number of
heartbeats over its lifetime if its body mass is 1.7 105 kilograms.
SOLUTION
about 7.92
108 heartbeats
WHAT OPERATIONS CAN
BE PERFORMED ON A PAIR
OF FUNCTIONS TO OBTAIN
A THIRD FUNCTION?
Two functions can be combined by the operations:
+, -, x, ÷ and composition
ESSENTIAL QUESTION
SOLVE X=4Y3 FOR Y
LESSON 4:
USE INVERSE FUNCTIONS
HOW DO YOU FIND AN
INVERSE RELATION OF A
GIVEN FUNCTION?
ESSENTIAL QUESTION
VOCABULARY
• Inverse relation: A relation that
interchanges the input and output
values of the original relation. The graph
of an inverse relation is a reflection of
the graph of the original relation, with
y=x as the line of reflection
• Inverse function: An inverse relation that
is a function. Functions f and g are
inverses provided that f(g(x)) = x and
g(f(x)) = x
EXAMPLE 1
Find an inverse relation
Find an equation for the inverse of the relation y = 3x – 5.
y = 3x – 5
x = 3y – 5
x + 5 = 3y
1
5
x+
=y
3
3
Write original relation.
Switch x and y.
Add 5 to each side.
Solve for y. This is the inverse
relation.
Verify that functions are inverses
EXAMPLE 2
Verify that f(x) = 3x – 5 and f
are inverse functions.
–1(x)
5
1
x+
3
3
=
SOLUTION
STEP 1
STEP 2
Show: that f(f –1(x)) = x.
Show: that f –1(f(x)) = x.
f (f
–1(x))
=f
=3
1
x+
3
5
3
1
x+
3
5
3
f –1(f(x)) =
–5
f –1((3x – 5)
=
1
(3x – 5) +
3
=x+5–5
=x–
=x
=x
5
5
+
3
3
5
3
for Examples 1, 2, and 3
GUIDED PRACTICE
Find the inverse of the given function. Then verify that your result and
the original function are inverses.
1.
f(x) = x + 4
SOLUTION
2.
3.
x1
x–4 =y
SOLUTION
f(x) = 2x – 1
SOLUTION
x+1
2
f(x) = –3x – 1
=y
3
=y
for Examples 1, 2, and 3
GUIDED PRACTICE
4.
Fitness: Use the inverse function in Example 3 to find the length at
which the band provides 13 pounds of resistance.
SOLUTION
48 inches
HOW DO YOU FIND AN
INVERSE RELATION OF A
GIVEN FUNCTION?
Write the original equation.
Switch x and y.
Solve for y.
ESSENTIAL QUESTION
EXPAND AND SOLVE:
(X-5)2
LESSON 6:
SOLVE RADICAL EQUATIONS
WHY IS IT NECESSARY TO
CHECK EVERY APPARENT
SOLUTION OF A RADICAL
EQUATION IN THE ORIGINAL
EQUATION?
ESSENTIAL QUESTION
VOCABULARY
• Radical equation: An equation with one
or more radicals that have variables in
their radicands
• Extraneous solution: An apparent
solution that must be rejects because it
does not satisfy the original equation.
Solve a radical equation
EXAMPLE 1
Solve 3  2x+7= 3.
3
 2x+7
( 3 2x+7 )3
=3
Write original equation.
= 33
Cube each side to eliminate the radical.
2x+7 = 27
2x = 20
x = 10
Simplify.
Subtract 7 from each side.
Divide each side by 2.
Solve a radical equation
EXAMPLE 1
CHECK
Check x = 10 in the original equation.
3
 2(10)+7
3
 27
3
=? 3
Substitute 10 for x.
=? 3
Simplify.
= 3
Solution checks.
GUIDED PRACTICE
for Example 1
Solve equation. Check your solution.
1.
3√
x – 9 = –1
ANSWER
2.
3.
x = 512
( x+25 ) = 4
ANSWER
x = –9
(23 x – 3 ) = 4
ANSWER
x = 11
WHY IS IT NECESSARY TO
CHECK EVERY APPARENT
SOLUTION OF A RADICAL
EQUATION IN THE ORIGINAL
EQUATION?
Raising both sides of an equation to the same
power sometimes results in an extraneous solution
ESSENTIAL QUESTION