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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
2
Simplify a .
b
a2
b2
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.
Example: 8 = 2 • 2 • 2 = 4
2•5
5
10
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
√a
a
numbers,
=
.
√b
b
Example 8
25
Simplify
.
4
25 = √25 = 5
2
4
√4
Example 9
√45
Simplify
.
√5
√45 =
√5
45 = √ 9 = 3
5
Example 10
√18
Simplify
.
√9
√18 =
√9
18 = √ 2
9
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