Section 1-8 Radical Rules
PRODUCT RULE:
𝑎
𝑎
QUOTIENT RULE:
𝑎
√𝑥
√𝑦
𝑎
𝑥
𝑎
√𝑥 ∙ √𝑦 = √𝑥𝑦
𝑎
= √
𝑦
Example:
Example:
√10 ∙ √𝑥 = √10𝑥
√10
√2
=√
10
2
= √5
More directly, when determining a product or quotient of radicals and the indices (the small number in front
of the radical) are the same then you can rewrite 2 radicals as 1 or 1 radical as 2.
Simplify by rewriting the following using only one radical sign (i.e. rewriting 2 radicals as 1).
1. √3 ∙ √12
2.
√12
√3
3
3.
√7𝑥 ∙ √2𝑦
4.
√12𝑥 2
3
√4𝑥
Simplify by rewriting the following using multiple radical sign (i.e. rewriting 1 radical as 2).
144
5. √ 25
𝑥6
6. √121
M. Winking (Section 1-8)
p. 17
Express each radical in simplified form.
8. √450𝑥 4 𝑦 5
7. √48
9. √72𝑥 5 𝑦 6
P
10. √300𝑥 12
N
11. √675𝑥 4 𝑦 11
12. −√81𝑥 3 𝑦 8
O
3
13. √48𝑥 7 𝑦 3
O
N
14.
A
3
3
√81𝑥10 𝑦 3
15. √−27𝑥 5
N
I
K
Use the letters and answers to match the answer to the riddle. Only some answers will be used.
“What is an opinion without π?”
_________
−9𝑥𝑦 4 √𝑥
_________
15𝑥 2 𝑦 2 √2𝑦
_________
10𝑥 6 √3
_________
3
2𝑥 2 𝑦 √6𝑥
_________
3
3𝑥 3 𝑦 √3𝑥
_________
_________
6𝑥 2 𝑦 3 √2𝑥
15𝑥 2 𝑦 5 √3𝑦
M. Winking (Section 1-8)
p. 18
Express each radical in simplified form.
17. √18𝑎5 ∙ √6𝑎4
16. √6𝑥 ∙ √12𝑥
Simplify. Assume that all variable represent positive real numbers.
18. 5√3 + √2 − 2√3 + 4√2
19.
21. 𝑦√18 − 3√12𝑦 4 + 2√8𝑦 2

24. 2 10 3 6  2 5  24

108  5 12  4 44
23. 5 18x 4  3x 8x 2  x 2 2
22. 𝑥 √32𝑥 2 + 2 √18𝑥 4
25.
2x

6x  3 x
20. 2 150  18  3 8  24

26. 3 6a

4a  2 15a 2

p. 101
M. Winking (Section 1-8)
p. 19
Simplify. Assume that all variable represent positive real numbers and rationalize all denominators.
18.
3
√5
16
21. √27
19.
6
√3
22.
20.
12  8 3  2 27
2
3√2
√6
23.
√2(√12−√3)
√3
M. Winking (Section 1-8)
p. 20