Download Math 137 Intermediate Algebra

Math 110 Modeling with Elementary Functions
Worksheet # 12
Fall 2004
V. J. Motto
Name _________________________________________
Roots and Radicals
The square root of a number is defined as one of its two equal factors. For example, 5
is the square root of 25 because 5 is one of the two equal factors of 25. In a similar way,
a cube root of a number is one of its three equal factors.
Number
9 = 32
Equal Factors
3▪3
Root
3
Type
Square root
25 = (-5)2
(-5) ▪ (-5)
-5
Square root
-27 = (-3)3
(-3) ▪(- 3)
-3
Cube root
16 = 24
2▪2▪2▪2
2
Fourth root
Definition of n’th Root of a Number
Let a and b be real numbers and let n be an integer such that n ≥ 2. If
a = bn
then b is the n’th root of a. If n = 2, the root is a square root, and if n = 3, the root is a
cube root.
Some numbers have more than one n’th root. For example, both 5 and -5 are square root
of 25 because 25 = 52 and 25 = (-5)2. To avoid ambiguity about which a number are
taking about, the principle n’th root of a number is defined in terms of a radical symbol
n
.
Principal n’th Root of a Number
Let a be a real number that has at least one (real number) n’th root. The principal n’th
root of a is the n’th root that has the same sign as a, and it is denoted by the radical
symbol
n
a.
1
The positive integer n is the index of the radical, and the number a is the radicand. If n
= 2, omit the index and write √ a rather than 2√a.
Using the radical symbol √ to write that the square roots of the following numbers is
desired. Then find the square root.
1. 36
2.
81
3.
49
4.
144
Using the radical symbol 3√ to write that the cube roots of the following numbers is
desired. Then find the cube root.
5.
27
6.
-27
7. 125
8. -8
Properties of the n’th Powers and n’th Roots
Let a be a real number, and let n be an integer such that n ≥ 2.
1. If a has a principal n’th root, then ( n a )n = a.
2.
Assume that a < 0.
a. If n is odd and a < 0, then
n
a n is a.
b. If n is even ( n a )n = | a | .
Evaluate each radical expression.
9.
11.
3
53
_________________
( 3) 2
______________
10.
12.
2
3
( 2)3
32
____________________
____________________
Definition of Rational Exponents
Let a be a real number, and let n be an integer such that n ≥ 2. If the principal n’th root
of a exists, we define a 1/n to be
a 1/n = n a
If m is a positive integer that has no common factor with n, then
a m/n = (a 1/n)m = ( n a ) m
and a m/n = (a m) 1/n =
n
am
Evaluate the following expressions with Rational Exponents.
13. 8 4/3 ___________________
15. 25
-3/2
14. (42)3/2 __________________
64
16. (-----)2/3 __________________
12
__________________
17. – 9 1/2 __________________
18. (-9)1/2 __________________
Rewrite expression using rational exponents.
19. x
4
x3
______________________________________
3
x2
20. ----------x
______________________________________
3
Simplify.
21.
3
x
_______________________________________
(2x – 1) 4/3
22. ----------------3
2x 1
_______________________________________
3