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