7.6 Properties of Radicals
The symbol n b is called a
b is called the
n is called the
n
b n  b if n is
.
.
.
, and
We can take an
negative number.
n
b n  b if n is
.
root of a negative number, but we cannot take an
root of a
Questions 1-3: State the value of the root, or state that the expression is not defined as a real number.
1.
3
Theorem:
For all, a, b,
1.
2.
n
n
2.
125
n
a , and
n
4
3.
81
3
64
b:
ab  n a  n b
a na

b nb
b  0 
This theorem works in reverse as well. We can use it to simplify radicals or to combine two radicals. Just
remember that in order to combine two radicals they MUST have the same index!
Questions 4-6: Simplify. (use perfect squares, cubes, etc)
4.
320
Remember, we never want a
5.
3
6.
40
in the denominator. So, we need to rationalize it!
Questions 7-8: Simplify.
7.
121
3
125
64
8.
4
36
25
Theorem:
For all b and
n
n
bm 
b , and m and n positive integers,
 b
m
n
9. Simplify.
4
812
Theorem:
For k and m integers and all b and
km
b
k m
10. Simplify.
8
km
b,
b mkb
36
Simplifying Roots with Variables
To simplify a root with variables we divide the exponent on the variable by the index. The quotient is the
exponent on the variable
the radical, and the remainder is the exponent on the
variable
the radical.
Questions 11-12: Simplify.
11.
3
12.
250x 8
192a11
Questions 13-18: Simplify.
13.
3
12b
14.
3
375  3 72
15.
3
320
50
16.
75a12 b 5
17.
5
32x10 y15
18.
3
24ab 6
z2