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 ___________.
a < 0 One real nth root: n a  a1/n
Ex) x3  8, then ___________.
a = 0 One real nth root: n 0  0
Ex) x 2  0, then ___________.
a = 0 One real nth root: n 0  0
Ex) x3  0, then ___________.
a > 0 Two real nth roots:  n a  a1/n
Ex) x 2  5, then ___________.
a > 0 One real nth root: 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 1: 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
1
 an
a m / n   a1/ n  
m
 a
n
m
a m/ n 
1
a
Example 2: Evaluate:
a) 1252/3
b) 84/3
You Try 2: Evaluate.
a) 45/2
b) 9 1/ 2
m/ n

1
 a1/ n 
m

1
 
n
(a  0)
m
a
Example 3: Evaluate the expression using a calculator. Round the result to two decimal places when
appropriate.
a) 221/4
Algebra 2_Ch.6A Notes-page1
b) 355/6
c)
 11 
5
4
You Try 3: Evaluate the expression using a calculator. Round the result to two decimal places when
appropriate.
b) 64 2/3
a) 42/5
Example 4: Solve each equation.
a) 6 x 3  384
c)
b)
 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
Algebra 2_Ch.6A Notes-page2
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
a 
m
 ab 
n
 a mn
m
 a mb m
am 
1
,a  0
am
am
 a mn , a  0
an
m
am
a

,b  0
 
bm
b
Example 1: Use the properties of rational exponents to simplify the expression.
5
a. 121/8 125/6
b.
10
d.
102/5
 561/4 
e.  1/4 
 7 
1/3
 71/4

3
c.
2
6
 46

5
You Try: Simplify.
3/4
a. 2
2
1/2
Algebra 2_Ch.6A Notes-page3
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 largest
10
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
Algebra 2_Ch.6A Notes-page4
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
Algebra 2_Ch.6A Notes-page5
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
Algebra 2_Ch.6A Notes-page6