1. 10-1 Simplifying Radicals
Hubarth
Algebra

2. Radical Expressions like 2 3 and 𝑥+3 contain a radical.
Multiplication Property of Square Roots
For every number 𝑎≥0 𝑎𝑛𝑑 𝑏≤0, 𝑎𝑏 = 𝑎 ∙ 𝑏 .
EXAMPLE = 9 ∙ 6 =3∙ 6 =3 6

3. Ex 1 Simplifying Square Roots
a b c
9 ∙ 5
16 ∙ 3
81 ∙ 3
3 ∙ 5
3 5
4∙ 3
4 3
9∙ 3
9 3

4. Ex 2 Remove Variable Factors
Simplify
𝑎 𝑥 b 𝑥 c 𝑎 12
9 𝑥 4 ∙ 3𝑥
4 𝑥 6 ∙ 7𝑥
4 𝑎 6
3𝑥 2 3𝑥
2𝑥 3 7𝑥

5. Ex 3 Multiplying Two Radicals
Simplify each radical expression.
a •
b x • x
12∙32
𝑥 2
384
21 4 𝑥 2 ∙ 10
64 ∙ 6
21∙2𝑥∙ 10
8 6
42𝑥 10

6. Division Property of Square Roots
For every number 𝑎≥0 𝑎𝑛𝑑 𝑏>0, 𝑎 𝑏 = 𝑎 𝑏
EXAMPLE = = 4 5

7. Ex 4 Simplifying Fractions Within Radicals
Simplify each radical expression. a. 13 64 b. 49 x4 = 13 64 = 13 8 = 49 x4 7 x2 =

8. Ex 5 Simplifying Radicals by Dividing
Simplify each radical expression. a.
120 10 b.
75x5
48x
120 10 = =
75x5
48x
25x4 16
= 4 • 3 =
25x4 16
= 4 • 3 =
25 • x4 16 =
5x2 4 =

9. Ex 6 Rationalizing a Denominator
Simplify by rationalizing the denominator. a. 3 7 b. 11
12x3 3 7
√ 7
= • √ 3x 11
12x3
= •
3 7 49 =
33x
36x4 =
3 7 7 =
33x
6x2 =

10. Practice
1. Simplify each radical expression.
a b 𝑥 9 𝑦 c 𝑎 2 ∙6 10 𝑎 3
d e 𝑝 3 𝑞 2
2. Rationalize the denominator.
a b
5 2
3 𝑥 4 𝑦 5 6𝑥
60 𝑎 2 2𝑎 4
5𝑝 𝑝 𝑞 3
2 6