6.1 – Rational ExponentsRadical Expressions

Finding a root of a number is the inverse operation of raising a number to a power.

This symbol is the radical or the radical sign

n aindex

radical sign

radicand

The expression under the radical sign is the radicand.

The index defines the root to be taken.

Square Roots

If a is a positive number, then

a is the positive square root of a and

100

a is the negative square root of a.

A square root of any positive number has two roots – one is positive and the other is negative.

Examples:

10

25

49

5

7

11 36 6

9 non-real # 81.0 9.0

6.1 – Rational Exponents

Cube Roots

3 27

A cube root of any positive number is positive.

Examples:

3 5

43

125

64

3 8 2

A cube root of any negative number is negative.

3 a

3 27 3 3 8 2

6.1 – Rational Exponents

nth Roots

An nth root of any number a is a number whose nth power is a.

Examples:

2

4 81 3

4 16

5 32 2

43 81

42 16

52 32

6.1 – Rational Exponents

Examples:6x

6.1 – Rational Exponents

3x 18y 9y

1216216 cba 684 cab3 188y 62y

4 1216881 cba 3423 cba

The value of the numerator represents the power of the radicand.

Examples:

:nm

aofDefinition

The value of the denominator represents the index or root of the expression.

n ma or mn a

31

272521

25 35 3 27

3423

4 64 8

6.1 – Rational Exponents

3423

432 8

or

Examples:

:nm

aofDefinition n ma or mn a

72

12 x 7 212 x

6.1 – Rational Exponents

32

32

27

132

27

1

3 2

3 2

27

19

13

3

729

1

32

32

27

132

27

1

23

23

27

1

9

1 2

2

3

1

or

6.1 – Rational Exponents

More Examples:

:nm

aofDefinition n ma or mn a

43

43

81

1643

8116

34

34

81

16278

3

3

32

51

21 3225 7255 3225

6.1 – Rational Exponents

Examples:

:nm

aofDefinition

n ma

1

mn a

1

21

25

12

125

25

1

5

1

32

1

x3

2x 3 2

1

x 23

1

x

nma

1or or

or

32

2764

23

23

64

2732

6427

22

43

169

6.1 – Rational ExponentsUse the properties of exponents to simplify each expression

35

34xx 3

9x

3x

101

53 x

101

53

x

x10

110

6 x 10

5x

42

3x4 281x 21

3x

35

34 x

21x

3 212 xx 32

121 x 12

9x

43x

32

121xx 12

8121 x

6.1 – Rational ExponentsUse the properties of exponents to simplify each expression

72

35

34yx

3

2101

53

53

x

x

72

35

72

34 yx 21

1021

8yx

32

101

53

53

x

x151

259

x

x151

259 x

755

7527 x 75

22x