3.4 warm up factor the expressions. 1. x² + 8x + 7 2. x² - 7x + 10 3. x² + 2x - 48

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

Quotient Property of Radicals Square root of a quotient equals the

quotient of the square roots of the numerator and denominator

36

9 * 4

4

25

4

25

2

5

EXAMPLE 1 Use the product property of radicals

a. Factor using perfect square factor.

Product property of radicals

Simplify.

Factor using perfect square factors.

Product property of radicals

32 = 16 2

= 16 2

= 4 2

b. x39 = 9 x2 x

= 9 x2 x

= 3x x Simplify.

GUIDED PRACTICE for Example 1

a.

Simplify1.

24 = 2 6

b. x225 = 5x

EXAMPLE 2Multiply radicals: Anytime you have 2 of the same, you should “pull one out”

a. Product property of radicals

Simplify.

Product property of radicals

= 6 6

= 36

6

= 6

6

Simplify.

Multiply.

= x234

= x24 3

= 3x x4b. 3x 4 x

4x 3=

Multiply.

Product property of radicals

EXAMPLE 2 Multiply radicals

c. Product property of radicals

Simplify.

Product property of radicals

= 73xy

Multiply.

3 x7xy2 = 7xy2 x3

= 3 7x y22

= 7 x23 y2

EXAMPLE 3 Use the quotient property of radicals

a.

Simplify.

13100

= 13100

=1013

Quotient property of radicals

b. 7x2 = 7

x2

= x7 Simplify.

Quotient property of radicals

GUIDED PRACTICE for Examples 2 and 3

Simplify2.

xa. x32 = x 22

b. 1y2 = y

1

EXAMPLE 4 Rationalize the denominator: Multiply by 1

a.

Product property of radicals

Simplify.

75 =

75 7

7

=49

75

= 775

Multiply by .7

7

EXAMPLE 4 Rationalize the denominator: Multiply by 1.

b.

Product property of radicals

Simplify.

Product property of radicals

3b2 = 3b

3b3b2

=9b6b

2

=b29

6b

= 6b3b

Multiply by .3b3b

EXAMPLE 5 Add and subtract radicals: Must have same radical to combine terms.

a.

Simplify.

=

Simplify.

4

Product property of radicals

10 Commutative property13+ – 9 10 4 10 – 9 10 13+

= (4 – 9) 10 13+

–5 10 13+=

Distributive property

35b. + 48 16 3= 35 +

= 3(5 + 4)

= 35 + 16 3

= 35 34+

39=

Factor using perfect square factor.

Distributive property

Simplify.

GUIDED PRACTICE for Examples 4 and 5

3. 31 = 3

3

Simplify the expression.

4. x1

= xx

5. 32x

= 2x2x3

6. 72 + 633 711=

GUIDED PRACTICE for Examples 6 and 7

Simplify the expression7.

( )54 – ( )1 – 5 = 9 – 5 5

EXAMPLE 6 Multiply radical expressions

a.

Simplify.

(4 –5

Product property of radicals

20 Distributive property

Simplify.

) = 4 – 205 5

= 4 5 – 100

= – 104 5

b. + – 3( )( )2727

= 7 27 3– + 7 2 2– 3( )2

= 147– 3 + 14 – 6

14= 1 – 2

Multiply.

Product property of radicals

Simplify.

= ( ) ( ) (+ 2 + +–327 7 2 7 –3 )2 2

Rationalize the denominator:

1

5 + √3

2

5 - √3

EXAMPLE 7 Solve a real-world problem

ASTRONOMY

a. Simplify the formula.

b. Jupiter’s average distance from the sun is shown in the diagram. What is Jupiter’s orbital period?

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 where d is the average distance (in astronomical units, abbreviated AU) of the planet from the sun.

3

EXAMPLE 7 Solve a real-world problem

= 2d d

Product property of radicals

Simplify.

SOLUTION

= 3a. P d

= 2d d

= d d

Factor using perfect square factor.

Write formula.

b. Substitute 5.2 for d in the simplified formula.

= 5.2=P d d 5.2

The orbital period of Jupiter is 5.2 , or about 11.9, Earth years.

ANSWER

5.2