3.4 Warm Up
Factor the expressions.
1. x² + 8x + 7
2. x² - 7x + 10
3. x² + 2x - 48
3.4 Simplify Radical
Expressions
Simplest Form of a Radical
 No perfect squares in radicand
(other than1)
 No fractions in radicand
 No radicals in the denominator
 Rationalize
the denominator
Properties of Radicals
 Product Property of Radicals

Square root of the product equals the
product of the square roots of the factors
36
9* 4
 Quotient Property of Radicals
 Square root of a quotient equals the
quotient of the square roots of the
numerator and denominator
4
25
4
25
2
5
EXAMPLE 1
a.
32 =
=
Use the product property of radicals
16 2
Factor using perfect square factor.
16
2
Product property of radicals
= 4 2
b.
9 x3 =
=
Simplify.
9 x2 x
9
= 3x x
x2
Factor using perfect square factors.
x
Product property of radicals
Simplify.
GUIDED PRACTICE
1.
Simplify
= 2 6
a.
24
b.
25x2 = 5x
for Example 1
Multiply radicals: Anytime you have 2 of the
same, you should “pull one out”
EXAMPLE 2
a.
b.
6
6
3x 4 x
=
6 6
Product property of radicals
=
36
Multiply.
=
6
=
4 3x
Simplify.
x
Product property of radicals
= 4 3 x2
= 4
=
3
4x 3
Multiply.
x2
Product property of radicals
Simplify.
EXAMPLE 2
c.
Multiply radicals
7xy2 3 x = 3 7xy2 x
= 3 7x 2 y2
= 3
7
= 3xy 7
Product property of radicals
Multiply.
x2
y2 Product property of radicals
Simplify.
EXAMPLE 3
a.
b.
13
100
7
x2
Use the quotient property of radicals
=
13
100
Quotient property of radicals
=
13
10
Simplify.
=
7
x2
Quotient property of radicals
=
7
x
Simplify.
GUIDED PRACTICE
2.
for Examples 2 and 3
Simplify
a.
2 x3
b.
1
y2
x
=
2
= x
1
y
2
EXAMPLE 4
a.
5
7
Rationalize the denominator: Multiply by 1
=
5
7
=
5 7
49
5 7
=
7
7
7
Multiply by
7 .
7
Product property of radicals
Simplify.
EXAMPLE 4
b.
2
3b
Rationalize the denominator: Multiply by 1.
3b
3b
Multiply by
3b .
3b
=
2
3b
=
6b
9b2
Product property of radicals
=
6b
9
b2
Product property of radicals
=
6b
3b
Simplify.
EXAMPLE 5
a.
Add and subtract radicals: Must have same
radical to combine terms.
4 10 + 13 – 9 10 = 4 10 – 9 10 + 13 Commutative property
= (4 – 9) 10 + 13 Distributive property
= –5 10 + 13
b.
5 3 + 48 = 5 3 + 16 3
= 5 3 + 16
= 5 3 +4 3
= (5 + 4) 3
= 9 3
Simplify.
Factor using perfect
square factor.
3 Product property of radicals
Simplify.
Distributive property
Simplify.
GUIDED PRACTICE
for Examples 4 and 5
Simplify the expression.
3.
1
3
4.
1
x
5.
3
2x
6.
=
3
3
=
x
x
=
3 2x
2x
2 7 + 3 63
= 11 7
GUIDED PRACTICE
for Examples 6 and 7
7. Simplify the expression
(4 – 5 ) (1 – 5 ) = 9 – 5 5
EXAMPLE 6
Multiply radical expressions
a. 5 (4 – 20 ) = 4 5 – 5
= 4 5 –
20 Distributive property
100 Product property of radicals
= 4 5 – 10
b.
Simplify.
( 7 + 2 )( 7 – 3 2 )
2
= ( 7 ) + 7 (–3 2 ) + 2
7 + 2 (–3 2 )
2
Multiply.
= 7 – 3 7 2 + 7 2 – 3( 2 ) Product property of radicals
= 7 – 3 14 + 14 – 6
Simplify.
= 1 – 2 14
Simplify.
Rationalize the denominator:
1
5 + √3
2
5 - √3
EXAMPLE 7
Solve a real-world problem
ASTRONOMY
The orbital period of a planet is
the time that it takes the planet to
travel around the sun. You can
find the orbital period P (in Earth
years) using the formula P = d 3
where d is the average distance
(in astronomical units,
abbreviated AU) of the planet
from the sun.
a.
Simplify the formula.
b.
Jupiter’s average distance from the sun is shown
in the diagram. What is Jupiter’s orbital period?
EXAMPLE 7
Solve a real-world problem
SOLUTION
a.
P =
d3
Write formula.
= d2 d
=
d2
= d d
b.
Factor using perfect square factor.
d
Product property of radicals
Simplify.
Substitute 5.2 for d in the simplified formula.
P = d d = 5.2 5.2
ANSWER
The orbital period of Jupiter is 5.2 5.2 , or about 11.9,
Earth years.