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3.1 Evaluate nth Roots and
Use Rational Exponents
p. 166
What is a quick way to tell what kind of real
roots you have?
How do you write a radical in exponent form?
What buttons do you use on a calculator to
approximate a radical?
What is the difference between evaluating and
solving?
Real nth Roots
Let n be an integer greater than 1 and a be a real
number.
If n is odd, then a has one real nth root. n a  a1 n
If n is even and a > 0, then a has two real nth roots.
 a  a
n
1n
If n is even and a = 0, then a has one nth root.
n
0 0
1n
0
If n is even and a < o, then a has no real nth roots.
See page 166 for KEY CONCEPT
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
Find the indicated real nth root
n = 3, a = −125
n = 4, a = 16
3
 125 
3
 16 
4
3
 5
 5
4
2
 2
4
Rational Exponents
Let a1/n be an nth root of a, and let m be a
positive integer.
a
mn
a
    a
 a
m n

a
1n m
n
1
1
mn

a 
1n m
m

1
 a
n
m
,a  0
See page 167 for KEY CONCEPT
Evaluate (a) 163/2 and (b) 32–3/5.
SOLUTION
Rational Exponent Form
a. 163/2= (161/2)3= 43 = 64
b.
1
1
=
323/5 (321/5)3
1 = 1
= 3
8
2
32–3/5=
Radical Form
3
3/2
(
)
16 =  16 = 43 = 64
32–
3/5
1
1
= 3/5 = 5
32
(  32 )3
= 13 = 1
2
8
Evaluate the expression with
Rational Exponents
93/2
32-2/5
 9  3
3
1

25
32
3
1
 32 
5
2
 27

1

2
5
 5 2 


1 1
 2 
2
4
Approximate roots with a
calculator
Keystrokes
Expression
a.
91/5
b.
123/8
c.
( 4 7)3 = 73/4
Display
9
11
5 5
1.551845574
12
3 3
8
8
2.539176951
7
3 3
4
4
4.303517071
Using a calculator to
approximate a root
 5
4
3
 3.34
Rewrite the problem as 53/4 and enter
using ^ or yx key for the exponent.
Evaluate the expression using a
calculator. Round the result to two
decimal places when appropriate.
Keystrokes
Expression
9.
42/5
10.
64 –2/3
11.
(4√ 16)5
12.
(3√ –30)2
4
1
64
Display
2
5
1.74
-2
3
0.06
16
5
4
32
–30
2
3
9.65
Solve the equation using nth roots.
2x4 = 162
x4 = 81
x4 = 34
x = ±3
(x − 2)3 = 10
x  2  3 10
3
x  10  2
x ≈ 4.15
1 x5 = 512
2
SOLUTION
1 x5 = 512
2
x5 = 1024
x =
5
x = 4
1024
Multiply each side by 2.
take 5th root of each side.
Simplify.
( x – 2 )3 = –14
SOLUTION
( x – 2 )3= –14
(x–2)=
3
–14
x =
3
–14 + 2
x =
3
–14 + 2
x = – 0.41
Use a calculator.
( x + 5 )4 = 16
SOLUTION
( x + 5 )4 = 16
4
(x+5) =+
x =+
4
16
16 – 5
take 4th root of each side.
add 5 to each side.
x = 2 – 5 or
x = – 2 – 5
Write solutions separately.
x = –3
x = –7
Use a calculator.
or
Evaluating a model with roots.
When you take a number to with a rational
exponent and express it in an integer
answer, you have evaluated.
Solving an equation using an nth root.
When you have an equation with value that
has a rational exponent, you solve the
equation to find the value of the variable.
What is a quick way to tell what kind of real roots
you have?
Root is odd, 1 answer; root is even, 1 or 2 real
answers.
How do you write a radical in exponent form?
Use a fraction exponent (powers go up, roots go
down)
What buttons do you use on a calculator to
approximate a radical?
Root buttons
What is the difference between evaluating and
solving?
Evaluating simplifies; Solving finds answers x=.
Assignment
Page 169, 9-45 every 3rd problem, 50-56
even,
To get credit for doing the problem, you
must show the original problem along with
your answer unless it is a calculator
problem (41-51)