10.3 Simplifying Radical Expressions

Use the product rule for radicals.

Objective 1

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Product Rule for Radicals

If and are real numbers and n is a natural number, then

That is, the product of two nth roots is the nth root of the product.

n a n b.n n na b ab

Slide 10.3- 3

Use the product rule for radicals.

Use the product rule only when the radicals have the same index.

Multiply. Assume that all variables represent positive real numbers.

5 13 5 13

7 xy 7xy

65

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CLASSROOM EXAMPLE 1 Using the Product Rule

Solution:

Multiply. Assume that all variables represent positive real numbers.

3 32 7 3 2 7

66 2 38 2r r 6 516r

2 25 59 8y x xy 4 25 72y x

37 5

3 14

This expression cannot be simplified by using the product rule.

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CLASSROOM EXAMPLE 2 Using the Product Rule

Solution:

Use the quotient rule for radicals.

Objective 2

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Quotient Rule for Radicals

If and are real numbers, b ≠ 0, and n is a natural number,

then

That is, the nth root of a quotient is the quotient of the nth roots.

n a n b

.n

nn

a ab b

Slide 10.3- 7

Use the quotient rule for radicals.

Simplify. Assume that all variables represent positive real numbers.

10081

109

1125

115

8

16y

318125

3

3

18125

23

12

xr

3 2

3 12

x

r

3 185

8

16y

4

4y

3 2

4

xr

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CLASSROOM EXAMPLE 3 Using the Quotient Rule

Solution:

Simplify radicals.

Objective 3

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Conditions for a Simplified Radical

1. The radicand has no factor raised to a power greater than or equal to the index.

2. The radicand has no fractions.

3. No denominator contains a radical.

4. Exponents in the radicand and the index of the radical have greatest common factor 1.

Slide 10.3- 10

Simplify radicals.

Be careful with which factors belong outside the radical sign and which belong inside.

Simplify.

32 216 216 4 2

300 3100 3100 10 3

35 Cannot be simplified further.

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CLASSROOM EXAMPLE 4 Simplifying Roots of Numbers

Solution:

3 54 3 227 33 27 2 33 2

4 243 44 3 3 43 3

Simplify. Assume that all variables represent positive real numbers.

725p 3 225 ( )p p 35p p

372y x 2236 y y x 6 2y yx

7 5 63 27y x z3 6 3 2 63 3 y y x x z 2 2 233y xz yx

5 74 32a b4 4 4 34 2 2 a a b b 342 2ab ab

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CLASSROOM EXAMPLE 5 Simplifying Radicals Involving Variables

Solution:

Simplify. Assume that all variables represent positive real numbers.

12 32 1/1232 1/ 42

6 2t 1/62t 3 t

4 2

1/3t

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CLASSROOM EXAMPLE 6 Simplifying Radicals by Using Smaller Indexes

Solution:

If m is an integer, n and k are natural numbers, and all indicated roots exist, then

.kn nkm ma a

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Simplify radicals.

kn kma

Simplify products and quotients of radicals with different indexes.

Objective 4

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Simplify.

35 4

1/ 2 3/ 6 6 3 6 125 5 5 5

1/3 2/ 6 6 2 6 164 4 4

35 4 6 6125 16 6 2000

The indexes, 2 and 3, have a least common index of 6, use rational exponents to write each radical as a sixth root.

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CLASSROOM EXAMPLE 7 Multiplying Radicals with Different Indexes

Solution:

Use the Pythagorean theorem.

Objective 5

Slide 10.3- 17

Pythagorean TheoremIf c is the length of the longest side of a right triangle and a and b are lengths of the shorter sides, then

c2 = a2 + b2.

The longest side is the hypotenuse, and the two shorter sides are the legs, of the triangle. The hypotenuse is the side opposite the right angle.

Slide 10.3- 18

Use the Pythagorean theorem.

Find the length of the unknown side of the triangle.

c2 = a2 + b2

c2 = 142 + 82

c2 = 196 + 64c2 = 260

260c

4 65c

4 65c

2 65c The length of the hypotenuse is 2 65.

Slide 10.3- 19

CLASSROOM EXAMPLE 8 Using the Pythagorean Theorem

Solution:

Find the length of the unknown side of the triangle.

c2 = a2 + b2

62 = 42 + b2

36 = 16 + b2

20 = b2

20 b

4 5 b

4 5 b

2 5 bThe length of the leg is 2 5.

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CLASSROOM EXAMPLE 8 Using the Pythagorean Theorem (cont’d)

Solution:

Use the distance formula.

Objective 6

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Distance Formula

The distance between points (x1, y1) and (x2, y2) is

2 22 1 2 1( ) ( )d x x y y

Slide 10.3- 22

Use the distance formula.

Find the distance between each pair of points. (2, –1) and (5, 3)

Designate which points are (x1, y1) and (x2, y2).(x1, y1) = (2, –1) and (x2, y2) = (5, 3)

2 22 1 2 1( ) ( )d x x y y

2 2( ) (5 32 ( ))1d

2 2(3) (4)d

9 16d

5 52d

Start with the x-value and y-value of the same point.

Slide 10.3- 23

CLASSROOM EXAMPLE 9 Using the Distance Formula

Solution:

Find the distance between each pairs of points. (–3, 2) and (0, 4)

Designate which points are (x1, y1) and (x2, y2).(x1, y1) = (–3, 2) and (x2, y2) = (0, 4)

2 22 1 2 2( ) ( )d x x y y

2 2( ( )) (3 )0 4 2d

2 2(3) ( 6)d

9 36d

3 545d Slide 10.3- 24

CLASSROOM EXAMPLE 9 Using the Distance Formula (cont’d)

Solution: