8.1 Radical Expressions & Functions
The number c is a square root of a if c2 = a. The principal square root of a nonnegative number
is its nonnegative square root. The radical sign is used to indicate the principal square root of
the number over which it appears.
For each number find all of its square roots:
1. 64
2. 169
3. 3600
4. 256
Simplify:
5.
196
6.
 289

7.
49
121
8.
.0025
Identify the radicand, the root index, and the coefficient:
9.
3 b 2  5
10. 5 3
16
81
11.
3x 2y 3 6 x (x  y )
Find the specified value of each function if it exists:
12. f x   x 2  16
13.
m q   9  q2
14. g x   
3
2x  1
f  7  
m  7  
g 0  
f  2 
m  2 
g  62 
f 9 
m 9 
g  63  
Simplify, using absolute-value notation as needed:
15. p  z   3 
z3
64
16.
 4  q 2
17.
8
 228
14
18.
 6a 28
Simplify; assume no radicands were formed by raising negative quantities to even powers:
19.
 22c 2
20.
4
16x 4
21.
8
16
y2 y 
5
25
22.
3x  6
26.
x  7 18
Determine the domain of each function:
23.
x  7 
24.
6
3x  6
25.
3
20
x 20
8.2: Rational Exponents
n
If bn = a, then b is an nth root of a and b = a1/n =
, where a is called the radicand and n
is called the root index. If n is a positive even integer and a is a positive real number, then a1/n
is the positive real nth root of a and is called the principal root. If n is a positive odd integer and
a is any real number, then a1/n is the real nth root of a. If m and n are positive integers, then
am/n = (a1/n)m = (am)1/n, provided that a1/n is defined. . If m and n are positive integers, then
a -m/n = m1/ n , provided that a1/n is defined and nonzero. It is necessary to use absolute value
a
when taking even roots of variables to indicate the principal root. Product rule: n ab  n a n b .
a
Quotient rule:
n
a

b
n
n
a . A radical is simplified if it has (1) no perfect nth powers as factors of
b
the radicand (meaning no powers inside the radicand are larger than the root index), (2) no
fractions inside the radical, and (3) no radicals in the denominator.
Convert from radical to exponential or from exponential to radical form:
3
1. 3 27
2.
3.
4. -71/2
5. x-2/5
a3
w 27
Simplify (assume all variables represent non-negative real numbers)
6.
11.
7.
64
 1
 
 8
16. (w )
1
8. 1691/2
9. -161/4
13.
14.
10. (-16)1/4
3
9 1/3
21.
b 36
12. (-27)1/3
 243 
17. 

 32 
(a1/2b-1/3)4/3(a5b)
-3 5
36m
18. (16a8b4)1/4
22.
15.
242
19. 9-191/2
(9x8y-10z12)1/2
3
48
20. 32/392/3
23.
3
81a
1000
 144a 


18
 y

8
24.
96a 8b 7 c 6
26.
3
96a 8b 7 c 6
4
96a 8b 7 c 6
1
2
 32x
25.
 64x
10
-6
y
-15
y z
9
z


5
3
5
2
-12 - 3 =