1 7.1 and 7.2 roots and radical expressions and multiplying and dividing radical expressions

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7.1 and 7.2 Roots and Radical Expressions and Multiplying

and Dividing Radical Expressions

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Roots and Radical Expressions• Since 52 = 25, 5 is a square root of 25.

• Since (-5)2 = 25, -5 is a square root of 25.

• Since (5)3 = 125, 5 is a cube root of 125.

• Since (-5)2 = -125, -5 is a cube root of -125.

• Since (5)4 = 625, 5 is a fourth root of 625.

• Since (-5)4 = 625, -5 is a fourth root of 625.

• Since (5)5 = 3,125, 5 is a fifth root of 3,125.

And the pattern continues…….

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Roots and Radical Expressions• This pattern leads to the definition of the nth root.

• For any real numbers a and b, and any positive integer n, if an = b, then a is an nth root of b.

• Since 24 = 16 and (-2)4 = 16, both 2 and –2 are fourth roots of 16.

•Since there is no real number x such that x4 = -16, -16 has no real fourth root.

•Since –5 is the only real number whose cube is –125, -5 is the only real root of –125.

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Roots and Radical Expressions

Type of Number Number of Real nth Roots when n is Even

Number of Real nth Roots when n is Odd

Positive 2 1

0 1 1

Negative None 1

5

Finding All Real Roots• Find all real roots.

• The cube roots of 0.027, -125, 1/64

• The fourth roots of 625, -0.0016, 81/625

• The fifth roots of 0, -1, 32

• The square roots of 0.0001, -1, and 36/121

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Radicals• A radical sign is used to indicate a root.

• The number under the radical sign is called the radicand.

• The index gives you the degree of the root.

• When a number has two real roots, the positive root is called the principal root and the radicand sign indicates the principal root.

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Radicals• Find each real – number root.

3 274 81

49

4 16

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Radicals• Find the value of the expression of x = 5 and x = -5

2x

• For any negative real number a,

when n is even.aan n

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Radicals• Simplify each radical expression.

64x

3 63ba

109x

3 33ba

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Radicals• Simplify each radical expression.

4 416 yx

4 84 yx

424 yx

3 627c

4 128 yx

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Radicals are the inverse of exponents

2552

Exponents: Radicals:

525

12553 51253

62554 56254

312555 531255

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Simplify the Radicals

125 55

3 250 3 25

3 20003 3 230

5 640 5 202

3 2103

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Rules for Simplifying Radicals

• Square roots can simplify if there are sets of two duplicate factors.

• Cube roots can simplify if there are sets of three duplicate factors.

• Fourth roots can simplify if there are sets of four duplicate factors.

• Fifth roots can simplify if there are sets of five duplicate factors.

• And so on and so forth….

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7 7

7

14 28

14 14 2

14 2

15 3 12

5 3 3 4 3

3 2 5 3

6 15

8283 48 86

15


43 23

83

2

43 43

83 23

2 23

123 43 453

23 23 33 43 53 93

23 8

33 27

3 2 23 53

6 103

16


44 44

164

2

94 44 184 104

94 44 94 24 24 54

34 81

24 16

3 2 54

6 54

17


256x 8y 7z27

27 128

27 2 256

x 8 x 7 x

2xy 2xz27

256x 8y 7z23

4x 2y 2 4x 2yz23

23 23 22 256

x 8 x 3 x 3 x 2

y 7 y 3 y 3 y

18


243x18y 6z125

3x 3yz2 x 3yz25

256x 8y 7z23

4x 2y 2 4x 2yz23

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32x18y 6z12 2||4 639 zyx

10x 80y 34z1815 15 32525 10 zyxzyx

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32x12y 3z12

8x 5y 3z8

4x 7z4

128x15y13z85

2xy 2z65 5 2111464 zyx

xzx 23 ||2

Most of the time, it is easier to divide first, then simplify later.

5 2422 22 yzxyx

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64x 29y 31z106

16x15y136

4x14y18z106

x105y 32z2825

xy 2z625

x104y 30z2225 25 224464 zxzyx

6 4232 4|| zxzyx

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Rationalizing the Denominator

It is considered bad form to have a radical in the denominator of an expression.

It is necessary to do some algebra so that there is no longer a radical in the denominator.

2

8This should not be here.

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Rationalize the Denominator

2

8

To rationalize the denominator, you usually have to multiply by a fraction that is equal to one that also contains numbers that allow the offending radical to be removed.

Multiply by: 2

2

2

2

4

28

2

28 24

24


5

72 Multiply by:

5

5

5

5

25

352

5

352

5

72

25


Divide first

y

y

5

5

2

2

25

5

y

yx

||5

5 2

y

yx

xy

x

10

2 3

y

x

5

2

y

y

5

5Multiply by:

||5

5||

y

yx

26


3

3

6

4

x

Divide first

3

3

3

2

x Now multiply to rationalize the denominator.

3 2

3 2

)3(

)3(

x

x

3 3

3 2

27

18

x

x

x

x

3

183 2

27


3

3

5

12

x

Multiply to rationalize the denominator.3 2

3 2

)5(

)5(

x

x

3 3

3 2

125

300

x

x

x

x

5

3003 2

28


3 2

3

3

10

x

Multiply to rationalize the denominator.3

3

9

9

x

x

3 3

3

27

90

x

x

x

x

3

903

1 7.1 and 7.2 roots and radical expressions and multiplying and dividing radical expressions

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