Download 8x3 3√ p5q3

NAME
DATE
6-5
PERIOD
Study Guide and Intervention
Operations with Radical Expressions
Simplify Radicals
For any real numbers a and b, and any integer n > 1:
n
1. if n is even and a and b are both nonnegative, then √
ab =
n
n
n
ab = √
a √
b.
2. if n is odd, then √
Product Property of Radicals
n
√
a
n
√
b.
To simplify a square root, follow these steps:
1. Factor the radicand into as many squares as possible.
2. Use the Product Property to isolate the perfect squares.
3. Simplify each radical.
For any real numbers a and b ≠ 0, and any integer n > 1,
n
√
a
a
−
=−
, if all roots are defined.
n
b
√
b
Quotient Property of Radicals
√
n
To eliminate radicals from a denominator or fractions from a radicand, multiply the
numerator and denominator by a quantity so that the radicand has an exact root.
Example 1
3
Example 2
3
Simplify √
-6a5b7 .
3
3
3
2
8x3
−
.
5
√45y
2 3
8x
8x
−
= √−
√
45y
45y
3
5
3
Quotient Property
5
√
(2x)2 ․ 2x
= −
Factor into squares.
(3y2)2 ․ 5y
√
√
(2x)2 ․ √
2x
= −
(3y2)2 ․ √
5y
√
2⎪x⎥ √
2x
= −
3y2 √
5y
Product Property
Simplify.
5y
2⎪x⎥ √
2x √
= −․−
3y2 √
5y
5y
√
2⎪x⎥ √
10xy
= −
3
Rationalize the
denominator.
Simplify.
15y
Exercises
Simplify.
54
1. 5 √
4.
36
−
√
125
Chapter 6
4
9 20
2. √32a
b
5.
4 7
3. √75x
y
ab
−
√
98
6 3
6.
31
Lesson 6-5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
√
-16a b = √(-2)
2 a a (b ) b
3
2 √
= -2ab 2a2b
5 7
Simplify
pq
−
√
40
3
5 3
Glencoe Algebra 2
NAME
6-5
DATE
PERIOD
Study Guide and Intervention
(continued)
Operations with Radical Expressions
Operations with Radicals When you add expressions containing radicals, you can
add only like terms or like radical expressions. Two radical expressions are called like
radical expressions if both the indices and the radicands are alike.
To multiply radicals, use the Product and Quotient Properties. For products of the form
(a √b + c √
d ) ․ (e √f + g √
h ), use the FOIL method. To rationalize denominators, use
+ c √
conjugates. Numbers of the form a √b
d and a √
b - c √
d , where a, b, c, and d
are rational numbers, are called conjugates. The product of conjugates is always a
rational number.
Example 1
Simplify 2 √
50 + 4 √
500 - 6 √
125 .
50 + 4 √
500 - 6 √
125 =
2 √
=
=
=
Example 2
2
2 √
52 2 + 4 √
102 5 - 6 √5
5
2 5 √
2 + 4 10 √
5 - 6 5 √
5
10 √
2 + 40 √
5 - 30 √
5
10 √
2 + 10 √
5
Simplify (2 √
3 - 4 √
2 )( √
3 + 2 √
2 ).
(2 √3 - 4 √2 )( √3 + 2 √2 )
Simplify square roots.
Multiply.
Combine like radicals.
Example 3
2 - √5
3 + √
5
2 - √5
2 - √5
3 - √5
−=− −
√
√
√
3+ 5
3+ 5
3- 5
Simplify − .
- 3 √5
+ ( √5
)2
6 - 2 √5
)2
32 - ( √5
= −−
+5
6 - 5 √5
9-5
= −
11 - 5 √5
4
=−
Exercises
Simplify.
1. 3 √
2 + √
50 - 4 √
8
2. √
20 + √
125 - √
45
3
3
․ √
4. √81
24
5. √
2 √
4 + √
12
7. (2 + 3 √
7 )(4 + √
7)
8. (6 √
3 - 4 √
2 )(3 √
3 + √
2)
5 √
48 + √
75
5 √3
10. −
Chapter 6
3
(
3
3
)
3. √
300 - √
27 - √
75
6. 2 √
3 ( √
15 + √
60 )
9. (4 √
2 - 3 √
5 )(2 √
20 + 5)
5 - 3 √3
1 + 2 √3
4 + √2
2 - √2
11. −
12. −
32
Glencoe Algebra 2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
= 2 √
3 √
3 + 2 √
3 2 √
2 - 4 √
2 √
3 - 4 √
2 2 √
2
= 6 + 4 √
6 - 4 √
6 - 16
= -10
Factor using squares.