Section 8.6
Name: ________________________
Radical Expressions and Rational Exponents
Converting from exponential form to radical form.
a
b
x  b xa
Product Property of Radicals
Quotient Property of Radicals
x y  x  y
Note:
x

y
x2  y 2  x2  y 2
and
x
y
x2  y 2  x  y
*The denominator of the exponential form is called the root index when put into the radical form.
The root index of the radical shows the type of root. For example, a square root has an index of 2. A
1
 1

cube root has an index of 3.  x 2  x and x 3  3 x  .


2
1
1
2
1
1
5 3  3 52 or 3 5
Examples: 3 2  3
x2  1 
x
x2
 
2
3
Try these: a = _______
4
73 = _________
*The number or expression under the radical is called the radicand.
*Combining like terms can only be done if the radicands are the same. If so, add the coefficients of
the radicals, not the radicands:
x  x  1 x 1 x  2 x
Try these: 4 7  3 7 = _________
9 ab  2 ab = _________
*When adding or subtracting radicals, each radical must have the same root index and radicand.
Before combining like terms with radicals, simplify the radicals first to see if there are any like terms.
5 20  2 45  4 18
5 2  2 5  2 335  4 2  3 3
10 5  6 5  12 2
4 5  12 2
Simplify and combine like terms, if possible:
Try these: 3 72  5 18 = ________
6 3  2 27  48 = _____________
*The Product Property of Radicals is:
Examples:
Try these:
5
7  35
2 13
3 3  6 39
3
6 = ________
x
y  xy
6  12  4 3  2 3
2
m3  m6  m3
m3
6 5 3 5  18 25  90
x5
x3 = ___________
5 2 5 6 = _________
*Multiplying radicals with different indices:
3
a
a a
1
3
1
2
a a
2
6
3
6
a a
2 3

6 6
5
6
 a  6 a5
5
m 3 m = ___________
Try these: 7 3 7 = __________
__________________________________________________________________________
x
x
*The Quotient Property of Radicals is:

y
y
Examples:
6
6

5
5
10
10
10


16
4
16
*When writing radicals in simplified form, remove radicals from the denominator or remove fractions
from the radicand. To do this, you must rationalize the denominator. To rationalize the denominator,
multiply the numerator and denominator by a given quantity that ensures the denominator will become a
rational number or rational variable term.
5
5 3
15



3
3
3 3
Try this:
7
= __________
11
1.
Why is it not possible to combine
3
2  2 through addition?
A. They have different radicands.
C. They have different indices.
2. Multiply:
3
6
4
A.
B. They have the same radicands.
D. They have the same indices.
2 2
B.
6
2
C.
6
32
D.
6
45
3. Simplify 2 20 .
A. 4 5
B. 2 10
C. 8 5
D. 4 10
4. Which expression illustrates the Product Property of Radicals?
A.
n
C.
n
ab  n ab  n ab
a
ab  n
b
ab  n a  n b
n
a
n
D. ab 
n
b
B.
n
5. Which expression illustrates the Quotient Property of Radicals?
A.
n
C.
n
ab  n ab  n ab
a
ab  n
b
32 .
A. 2 4
ab  n a  n b
a na
n
D.

b nb
B.
n
6. Simplify
7. Simplify
B.
2
C. 16 2
D. 4 2
C. 24 48
D. 7 48
C. 5 2
D. 17 6
C. t  u
D. t u
 4 18  2 14 
A. 48 7
B. 24 24
98  72  32
A. 17 2
B. 5 6
8. Simplify
t4
u3
9. Simplify
A.
t2
u u
t2 u
B.
u2
2
2
y3
10. Simplify
A. y y
2
B. y
C.
y
D. y y
3
D. 3
3
81
3
9
11. Simplify
A.
3
9
B.
9
C.
3
45
12. Simplify
A. 3 5
B. 9 5
C. 5 3
D. 3 2
B. 8 2
C. 42
D. 6 2
B. 6461
C. 729
D. 243
B. 405
C. 9 10
D. 90
B. 13
C. 9
D. 11
128
13. Simplify
A. 64
5
14. Simplify 9
A. 59,049
810
A. 8 10
15. Simplify
16. Simplify
A. 8
4
6561
704
36
4 11
A.
3
17. Simplify
4 11
6
C.
B.
5 6
54
C. 54
D.
B.
2
4
1
8
D.
22  9
D. 4 11
50
972
18. Simplify
A. 5 6
6
9
72
19. Simplify
A.
4 11
B.
4
C.
1
2 2
Simplify the numerical or algebraic expression with radicals.
2
3
20.
23.
21. 6 3  5 3 
 4 18  2 14  
24.
26. 5 2  5 12  2 3 
29.
32.
34.
37.
3
5
8
2
3
 3

4 x 2  20 x  25 
25.
3
40  2 3 5 
27. 4 50  5 200 
28.
4
2 x 2 y  4 5 xy 
31.
5
2a 3  5 3a 
30.

9 x 2  24 x  16 

3 3 3 3 5 
8k 4 g 7 
18

72  2
90x 3 y 4 
22.
33.
35.
3


54 3 54 
3 7 5

7 2
38.
3 11  2 2
36.

11  1 
10

12  2