HOMEWORK and EXAM REVIEW on RADICALS
Part 1
When simplifying radical expressions, it is very useful to recognize perfect nth powers. The following
table might help.
12 
22 
32 
42 
52 
62 
72 
82 
92 
102 
112 
122 
13 
23 
33 
43 
53 
63 
14 
24 
34 
44 
15 
25 
Simplify each radical expression.
49
1.
16
3.
2.
3
48
4.
3
162
6.
108
8.
125
64
48
4
5.
7.
4
4
3
32
2
108
Evaluate the following expressions:
 3
9. (a)
11. (a)
4
 1
2
4
 3
(b)
(b)
4
1
2
4
10. (a)
3
 2 
3
12. (a)
5
 5
5
(b)
3
 2
(b)
5
 5
3
5
Note that we must be careful when evaluating radical expressions with an even index. If there are
variables under a radical, we must use caution in how we handle these variables because they might
represent negative values.
n
We have observed that
and
n
x n is not always equal to x. In fact
x n  x if n is ____________________
n
x n  x if n is ____________________.
The only time we have to consider using absolute value notation is when n is ____________________.
Simplify each radical expression where x represents any real number. Use absolute-value notation
wherever necessary.
13.
49x 2
14.
15.
x10
16.
17.
5
x10
18.
19.
3
64x3
20.
3
8x3
x4
3
x6
4
x16
Part II
Perform each indicated multiplication, and then simplify the product. Assume x > 0 and y > 0.
32 2
2 x 8x
1.
2.
3.
3
27x 3 x 2

5.
4 5 3 5 7
7.
8


7 1 2 7  3

4.
3 5  2 3 
6.
3 3 2 5 5 3
8.
5



y 3 x 5 y 3 x

Simplify each expression. Assume x > 0 and y > 0.
9.
11.
3
20x5
10.
16x 4 y8
12.
Conjugates Radicals: The conjugate of
x  y is
28x3 y 7
3
54x5 y10
x y.
Write the conjugate of each expression. Then multiply the expression by its conjugate.
Expression
Conjugate
Product
Example:
2 3
2 3

2 3


2  3  4  9  2  3  1
13. 3 5  2 7
14.
x 3 y
15.
2x  3 y
Perform the indicated divisions by rationalizing the denominator and then simplifying. Assume that all
variables represent positive real numbers.
16
8
16.
17.
7 3
3 5
18.
3
2 5
Find the sum or difference.
19.
4x
x y
21.
100  49
22. 5 4  2 9  4 25
23. 2 3 5  9 3 5
24. 3 5 x  7 5 x  4 5 x
25. 6 3 5  9 3 5  3 3 5
26. 3x y  4 x y  5x y
28. 3 xy  7 3 xy  4 xy  9 3 xy
27. 5 3 5  6 5  2 3 5  3 5
Find the sum or difference. Assume that all variables represent positive real numbers so that absolute
value notation is not necessary.
29.
45  20
30. 4 3 108 x3  5 3 32 x3
31.
3
3

16
4
32. 3
33.
28x3  3 700 x3
34. 2 3 40  3 3 135
35. 5 3 54 x2  2 3 128 x2
36.
5x
5x
4
25
49
4 x2  24 xy  36 y 2  x 2  y 2