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

Exercise Find the prime factorization of 144. 24 • 32

Exercise Simplify (xy)2. x2y2

Exercise ab 2 Simplify . a2b2

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

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

Simplified radicals have no perfect square factors in the radicand.

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

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

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

Example Simplify √ 2,500. 50

Example Simplify √ 360,000. 600

Example Simplify √ 49,000,000. 7,000

√ 40

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

Example Simplify √ 72. 6 √ 2

Example Simplify √ 240. 4 √ 15

Example Simplify √ 200. 10 √ 2

√ 180

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

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

√ 6 √ 15

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

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

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

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

Example Simplify √ 24 √ 54. 36

√ 36 √ 9 36 9

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

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

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

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

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

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

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

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