Radical Expressions
N2
N
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
N3
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
256
289
324
361
400
N4
1
8
27
64
125
216
343
512
729
1000
1331
1728
2197
2744
3375
4096
4913
5832
6859
8000
N5
1
16
81
256
625
1296
2401
4096
6561
10000
14641
20736
28561
38416
50625
65536
83521
104976
130321
160000
1
32
243
1024
3125
7776
16807
32768
59049
100000
161051
248832
371293
537824
759375
1048576
1419857
1889568
2476099
3200000
Square Root (Second Root)
0  00  0
1  12  1  1  1
4  22  2  2  2
9  32  3  3  3
16  42  4  4  4
25  52  5  5  5
36  62  6  6  6
49  7 2  7  7  7
64  82  8  8  8
81  92  9  9  9
100  102  10  10  10
121  112  11  11  11
144  122  12  12  12
169  132  13  13  13
196  142  14  14  14
225  152  15  15  15
256  162  16  16  16
289  17 2  17  17  17
324  182  18  18  18
361  192  19  19  19
400  202  20  20  20
1  Nonreal
4  Nonreal
9  Nonreal
16  Nonreal
25  Nonreal
36  Nonreal
49  Nonreal
64  Nonreal
81  Nonreal
100  Nonreal
Cube Root (Third Root)
3
0  3 03  3 0  0  0  0
3
1  3 13  3 1  1  1  1
3
8  3 23  3 2  2  2  2
3
27  3 33  3 3  3  3  3
3
64  3 43  3 4  4  4  4
3
125  3 53  3 5  5  5  5
3
216  3 63  3 6  6  6  6
3
343  3 73  3 7  7  7  7
3
512  3 83  3 8  8  8  8
3
729  3 93  3 9  9  9  9
3
1000  3 103  3 10  10  10  10
3
1  3 (1)3  3 (1)  (1)  (1)  1
3
8  3 (2)3  3 (2)  (2)  ( 2)  2
3
27  3 (3)3  3 (3)  (3)  (3)  3
3
64  3 (4)3  3 (4)  (4)  ( 4)  4
3
125  3 (5)3  3 (5)  ( 5)  ( 5)  5
3
216  3 (6)3  3 (6)  (6)  ( 6)  6
3
343  3 (7)3  3 (7)  ( 7)  (7)  7
3
512  3 (8)3  3 (8)  (8)  ( 8)  8
3
729  3 (9)3  3 (9)  (9)  ( 9)  9
3
1000  3 (10)3  3 (10)  ( 10)  ( 10)  10
Fourth Root
4
0  4 04  4 0  0  0  0  0
4
1  4 14  4 1  1  1  1  1
4
16  4 24  4 2  2  2  2  2
4
81  4 34  4 3  3  3  3  3
4
256  4 44  4 4  4  4  4  4
4
625  4 54  4 5  5  5  5  5
Example 1
Find the square of the expression 21.
Solution:
Square of 21 

21

2
 21  21  441  21
Example 2
Find the square of the expression - 12.
Solution:

Square of - 12  - 12
   - 12    - 12  
2
144  12
Example 3
Find the square of the expression 141.
Solution:
Square of 141 

 

2
141
141 

141  141
Example 4
Find the square of the expression
Solution:
Square of
x +21 
2

2
x +21

2
x 2 +21.
 x 2 +21  x 2 +21  x 2 +21
Classifying Numbers as Rational, Irrational, or
Nonreal
9 3
Rational
6  2.449489743
Irrational
-7  Nonreal
81  9
Rational
46  6.782329983
Irrational
-25  Nonreal
Finding Roots of Fractions
1
1 1

=
4
4 2
36
36 6

=
49
49 7
3
4
8

27
3
3
8

277
8 2

3
27
4
4
8
Nonreal
 4
 Nonreal
277
277
Assuming that x, y, and z are positive.
x 2 , y 2 , z 2 , ( x  1) 2 , (2 x  5) 2 , ( y  2 x) 2 , (3 x  yz ) 2
Simplify
Solution :
x2  x  x  x
y2 
y y  y
z2  z  z  z
( x  1) 2  ( x  1)( x  1)  x  1
(2 x  5) 2  (2 x  5)(2 x  5)  2 x  5
( y  2 x) 2  ( y  2 x)  ( y  2 x)  y  2 x
(3 x  yz ) 2  (3 x  yz )  (3 x  yz )  3 x  yz
Assuming that x, y, and z are positive.
Simplify 3 a 3 , 4 y 4 , 5 b5 , 3 (a  1)3 , 4 y 4 , 5 (b  5)5 , 3 ( ab  4) 3 ,
4
( y  z ) 4 , and
5
( ax  bz )5 .
Solution :
3
a3 
3
 a  a  a   a
4
y4 
4
 y  y  y  y   y
5
b5 
5
 b  b  b  b  b   b
3
(a  1)3  3 ( a  1)( a  1)( a  1)  a  1
4
(10 y  xz ) 4  4 (10 y  xz )(10 y  xz )(10 y  xz )(10 y  xz )  10 y  xz
5
(b  5)5  5 (b  5)(b  5)(b  5)(b  5)(b  5)  b  5
3
(ab  4)3  3 ( ab  4)( ab  4)( ab  4)  ab  4
4
( y  z ) 4  4 ( y  z )( y  z )( y  z )( y  z )  y  z
5
(ax  bz )5  5 (ax  bz )(ax  bz )(ax  bz )(ax  bz )(ax  bz )  ax  bz
Approximate the following with a calculator.
341  18.46618531
356  7.087341062
(Use a calculator and input expression as (-356)^(1/3))
3
1010  5.637419371
(Use a calculator and input expression as 1010^(1/3))
4
5687  5.636073468
(Use a calculator and input expression as (-5687)^(1/5))
5
Graphing Radical Functions
Graph f ( x)  x  2
x
2
3
6
11
f(x)
22 
3 2 
62 
11  2 
0 0
1 1
4 2
9 3
Point on graph
(2, 0)
(3, 1)
(6, 2)
(11, 3)
Graph f ( x)  x  5
x
-5
-4
-1
4
f(x)
5  5 
4  5 
1  5 
45 
0 0
1 1
4 2
9 3
Point on graph
(-5, 0)
(-4, 1)
(-1, 2)
(4, 3)