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3.4 Warm Up
Factor the expressions.1. x² + 8x + 7
2. x² - 7x + 10
3. x² + 2x - 48
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3.4 Simplify Radical Expressions
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Simplest Form of a Radical
No perfect squares in radicand (other than1)
No fractions in radicand
No radicals in the denominator Rationalize the denominator
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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
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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.
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GUIDED PRACTICE for Example 1
a.
Simplify1.
24 = 2 6
b. x225 = 5x
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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
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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
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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
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GUIDED PRACTICE for Examples 2 and 3
Simplify2.
xa. x32 = x 22
b. 1y2 = y
1
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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
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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
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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.
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GUIDED PRACTICE for Examples 4 and 5
3. 31 = 3
3
Simplify the expression.
4. x1
= xx
5. 32x
= 2x2x3
6. 72 + 633 711=
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GUIDED PRACTICE for Examples 6 and 7
Simplify the expression7.
( )54 – ( )1 – 5 = 9 – 5 5
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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
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Rationalize the denominator:
1
5 + √3
2
5 - √3
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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
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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