§2-4
RADICAL EXPRESSIONS
Definition
The nth root of a number a is a number that when raised to the nth power, produces the
is the radical and a is the
number a. It is written as n a , where n is the index,
radicand.
For example,
Similarly, 4 16 = 2 since 2 4 = 16 and
Definition
x 6 = x 3 since (x 3 )2 = x 6 .
9 = 3 since 32 = 9 and
3
x 12 = x 4 since (x 4 )3 = x12 .
m
, where m is the power of the number
n
m
and n represents the nth root of the number. Thus a m n = n a m = (n a ) where a ≥ 0.
Rational exponents are exponents of the form
23
13
For example, 27 = 3 27 = 3 and 8
= (3 8) = (2)2 = 4
2
Example 1
Write the expression x 2 3 in radical form.
Solution
The expression x 2 3 can be written as (x1 3 ) or (x 2 ) .
2
Therefore, x 2 3 can be written in radical form as
Example 2
13
( x ) or
3
2
3
2
x .
Simplify the expression 18 − 8 .
18 − 8 = 9⋅ 2 − 4 ⋅ 2 = 3 2 − 2 2 = (3 − 2) 2 = 2
Solution
Example 3
Simplify the expression
Solution
3
x 7 y5 z6 − 3 x10 y8 z 3 =
3
3
x 7 y5 z6 − 3 x10 y8 z 3 .
x 6 ⋅ x ⋅ y 3 ⋅ y 2 ⋅ z 6 − 3 x 9 ⋅ x ⋅ y6 ⋅ y 2 ⋅ z 3
= (x 2 yz2 )3 xy2 − (x 3 y2 z)3 xy2
= ( x 2 yz2 − x 3 y 2 z)3 xy2
= x 2 yz(z − xy)3 xy 2
(
2 − 3 )( 8 − 4 ).
Example 4
Multiply
Solution
( 2 − 3 )( 8 − 4 ) =
2 ⋅ 8 − 2 ⋅ 4 − 3 ⋅ 8 + 3⋅ 4
= 16 − 8 − 24 + 12
= 4 − 4 ⋅ 2 − 4 ⋅6 + 4 ⋅ 3
= 4−2 2 −2 6 +2 3
Copyright©2007 by Lawrence Perez and Patrick Quigley
§2-4
PROBLEM SET
Write each exponential expression as a radical expression.
1.
5
1
2
2.
(2x)
1
3
(xy3 )
3.
3
4
4.
− 5x
2
3
5.
(a 2 b 4 )
1
5
Write each radical expression as a exponential expression.
6.
11
7.
5
a
3
8.
− 2y
9.
4x x
10.
2 r 4 2s 3
4
9
Simplify each radical expression.
11.
64
12.
16.
x 12
17.
3
64
− a12b 4
13.
3
−1
14.
3
−27
15.
18.
3
8x 6 y 9
19.
4
16 r8
20.
−27a 9
22.
5
243x15 y 25
23.
25.
(x + y)2
26.
3
(x + y)6
27.
29.
2 12 − 3 3
30.
a 7−b 7
21.
3
3 5+2 5
28.
3 5−2 5
31.
3 ⋅ 3 16 − 3 2
32.
x − 2x x
3 2⋅ 3
36.
x+y⋅ x+y
40.
2x
8x 3
34.
x⋅ x
35.
37.
6x ⋅ 12x
38.
2
2
39.
45.
64
5 x 25y
⋅
y2
x3
42.
46.
3
x
43.
3y
x
⋅
4 4x y y
27a12
b15
3
x+ x
3
−x 5 y10
24.
33.
41.
25x 2
y4
5
47.
x
x
3
x 12
44.
5 x
25y
48.
2 ÷
y
x3
3
3 4
ab2
3y
x
÷
4 4x y y
49.
3
4
+
2
2
50.
3
4
−
2
2
51.
27
2
+3
12
27
52.
2
3
−
12
3
53.
2
3
+
12
3
54.
27
2
−3
27
12
55.
2 ( 10 + 5)
56.
3( 8 − 6)
57.
2
3− 2
58.
2
3+ 2
59.
3
1
+
60.
3+ 2
3− 2
2
3
−
2+ 5
2− 5
Copyright©2007 by Lawrence Perez and Patrick Quigley
§2-4
PROBLEM SOLUTIONS
1.
5
2.
3
2x
3.
y 2 ⋅ 4 x 3 y 4.
−5⋅ x
6.
111 2
7.
a
35
8.
−(2y)1 2
9.
4x
13.
-1
14.
-3
15.
11. 8
12. 4
3
32
2
5.
5
2
10.
(2 7 4 )(r)(s 3 4 )
a b
2
3
− xy2
17.
−a b
2
18.
2x 2 y 3
19.
2 r2
20.
22.
3x 3 y 5
23.
5x
y2
24.
3a 4
b5
25. x + y
(x + y)2
27.
5 5
28.
5
29.
3
5 ⋅3 2
32.
−x x
33.
2 x
36. x + y
37.
2x 18
38.
2
39.
41. 2
42.
6
x
43.
x
2
44.
47.
x2 y
y3
48.
y 2 3x
8x 2
49.
7 2
2
50.
52.
−
2 3
3
53.
4 3
3
54.
5
6
55.
6
16.
x
21.
− 3a
26.
31.
46.
3x
8y
3
6
51.
13
6
56.
2 6−3 2
57.
2 3+2 2
59.
4 3−2 2
60.
2 +5 5
3
34. x
x
24
ab
58.
2
4
30.
(a − b) 7
35.
3 6
40.
2x
2x
45.
25 y
xy2
2
2
2 5 + 10
2 3−2 2
Copyright©2007 by Lawrence Perez and Patrick Quigley