1. Exercise
Find the prime factorization of 700.
22 • 52 • 7

2. Exercise
Find the prime factorization of 144.
24 • 32

3. Exercise
Simplify (xy)2.
x2y2

4. Exercise ab 2
Simplify
a2b2

5. Exercise List the first twelve perfect squares.
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144

6. Fractions are simplified to lowest terms—no common factors in the numerator and denominator.
8 10
2 • 2 • 2 2 • 5 =
4 5 =
Example:

7. Simplified radicals have no perfect square factors in the radicand.

8. √ 4 • √ 9
= 2 • 3
= 6
but
√ 4 • √ 9
= √ 36
= 6

9. Product Law for Square Roots
For all a ≥ 0 and b ≥ 0, √ a • √ b = √ ab.

10. Example 1
Find √ 900.
√ 900
= √ 9 × 100
= √ 9 √ 100
= 3 × 10
= 30

11. Example
Simplify √ 2,500. 50

12. Example
Simplify √ 360,000.
600

13. Example
Simplify √ 49,000,000.
7,000

14. √ 40

15. Example 2
Simplify √ 48.
√ 48
= √ 3 × 16
= √ 3 √ 16
= 4 √ 3

16. Example
Simplify √ 72.
6 √ 2

17. Example
Simplify √ 240.
4 √ 15

18. Example
Simplify √ 200.
10 √ 2

19. √ 180

20. Example 3 Simplify √ 162. √ 162 = √ 2 • 3 • 3 • 3 • 3
= √ 2 √ 3 • 3 √ 3 • 3
= √ 2 • 3 • 3
= 9 √ 2

21. Example 4 Simplify √ 675. √ 675 = √ 3 • 3 • 3 • 5 • 5
= √ 3 √ 3 • 3 √ 5 • 5
= √ 3 • 3 • 5
= 15 √ 3

22. √ 6 √ 15

23. Example 5
Simplify √ 7 √ 3.
√ 7 √ 3
= √ 7 • 3
= √ 21

24. Example 6 Simplify √ 14 √ 21. √ 14 √ 21 = √ 2 • 7 √ 3 • 7
= √ 2 • 3 • 72
= √ 72 √ 2 • 3
= 7 √ 6

25. By the Associative and Commutative Properties, a √ b • c √ d = ac √ bd.

26. Example 7 Simplify 2 √ 54 • 3 √ 15. 2 √ 54 • 3 √ 15 = 6 √ 54 • 15
= 6 √ 2 • 3 • 3 • 3 • 3 • 5
= 6 • 3 • 3 √ 2 • 5
= 54 √ 10

27. Example
Simplify √ 24 √ 54. 36

28. √ 36
√ 9
36 9

29. Quotient Law for Square Roots
If a and b are positive real numbers, =
√a √b
a b

30. Example 8
25 4
Simplify
25 4 =
√25 √4 =
5 2

31. Example 9
√45 √5
Simplify
√45 √5 =
45 5
= √ 9
= 3

32. Example 10
√18 √9
Simplify
√18 √9 =
18 9
= √ 2

33. Exercise
Simplify √ x2y2. Assume that x ≥ 0 and y ≥ 0. xy

34. Exercise
Simplify √ x3y4. Assume that x ≥ 0 and y ≥ 0.
xy2√ x

35. Exercise
Under what circumstances is √ x2 = x true?
x ≥ 0

36. Exercise
Under what circumstances is √ x2 ≠ x true?
x < 0