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Name
Class
6-1
Date
Notes
Roots and Radical Expressions
For any real numbers a and b and any positive integer n, if a raised to the nth power
equals b, then a is an nth root of b. Use the radical sign to write a root. The
following expressions are equivalent:
What are the real-number roots of each radical expression?
a. 3 343
Because (7)3  343 , 7 is a third (cube) root of 343.
Therefore, 3 343  7 .
(Notice that (7)3  343 , so −7 is not a cube root of 343.)
4
4
1
b. 4
625
1
1
1
1
 1
1
Because   
and    
, both
and  are real-number
625
5
5
 5
 5  625
1
fourth roots of
.
625
c. 3 0.064
3
Because ( 0.4)  0.064 , −0.4 is a cube root of −0.064 and is, in fact, the
only one. So, 3 0.064  0.4 .
d.
25
Because (5)2 = (−5)2 = 25, neither 5 nor −5 are second (square) roots of −25.
There are no real-number square roots of –25.
Exercises
Find the real-number roots of each radical expression.
1. 169
2. 3 729
3. 4 0.0016
1
4. 3 
8
5.
4
121
125
6. 3
216
Name
Class
6-1
Date
Notes (continued)
Roots and Radical Expressions
You cannot assume that n a n  a . For example, (6)2  36  6 , not −6.
This leads to the following property for any real number a:
n an  a
If n is odd
n an  a
If n is even
What is the simplified form of each radical expression?
a. 3 1000x3 y9
3 1000 x3 y9  3 103 x3 ( y3 )3
b. 4
256 g 8
h 4 k16
Write each factor as a cube.
 3 (10 xy3 )3
Write as the cube of a product.
 10xy3
Simplify.
8
4 2 4
4 256 g  4 4 ( g ) Write each factor as a power of 4.
h4k16
h 4 (k 4 )4
 4g 2 

4
 hk 4 



4g 2
h k4
Write as the fourth power of a quotient.
Simplify.
The absolute value symbols are needed to ensure the root is positive when
h is negative. Note that 4g2 and k4 are never negative.
Exercises
Simplify each radical expression. Use absolute value symbols when needed.
1
7. 36x 2
8. 3 216y3
9.
100x 2
10.
x 20
y8
11. 3
( x  3)3
( x  4)6
12. 5 x10 y15 z5