Section 10.5

Rational Exponents and Radicals

10.5 Lecture Guide: Rational Exponents and Radicals

Objective 1: Interpret and use rational exponents.

Principal nth Root

For 0 :x

The principal nth root is positive for all natural numbers n.

For 0 :x

The principal nth root of 0 is 0.

For 0 :x

If n is odd the principal nth root is negative.

Examples inRadical Notation

16

3 27

0 0

3 0 0

3 64 5 32

The principal nth root of the real number x is denoted by either 1nx or .n x Examples in

Exponential Notation Verbally

1216

1327

120 0

130 0

1

364

1

532

Principal nth Root

Examples inRadical Notation

The principal nth root of the real number x is denoted by either 1nx or .n x Examples in

Exponential Notation Verbally

If n is even, there is no real nth root. (The nth roots will be imaginary.)

For 0 :x

1

a real number. is not

121

real number. is not a

Use the product rule for exponents to simplify the following expressions:

1.1 1

2 2x x

12x is the principal _________________________ root of x.

Use the product rule for exponents to simplify the following expressions:

2.1 1 1

3 3 3x x x

13x is the principal _________________________ root of x.

Use the product rule for exponents to simplify the following expressions:

3.1 1 1 1

4 4 4 4x x x x

14x is the principal _________________________ root of x.

4. 81

Represent each expression by using exponential notation, and evaluate each expression.

5.

Represent each expression by using exponential notation, and evaluate each expression.

4 81

6.

Represent each expression by using exponential notation, and evaluate each expression.

7 1

7.

Represent each expression by using exponential notation, and evaluate each expression.

3 8

Represent each expression by using radical notation, and evaluate each expression.

8.1

216

9.1

38

Represent each expression by using radical notation, and evaluate each expression.

10.1

416

Represent each expression by using radical notation, and evaluate each expression.

11.1

249

Represent each expression by using radical notation, and evaluate each expression.

12. 1

249

Represent each expression by using radical notation, and evaluate each expression.

13. 1

3125

Represent each expression by using radical notation, and evaluate each expression.

Examples inRadical Notation

Examples in Exponential Notation Algebraically

For a real number x and natural numbers m and n :

If 1nx

is a real number*

1 m mmnn nx x x

or

1m nm mnnx x x

2233

2

8 8

2

4

22 13 3

2

8 8

2

4

Rational Exponents

3377y y

3 137 7y y

Examples inRadical Notation

Examples in Exponential Notation Algebraically

For a real number x and natural numbers m and n :

If 1nx

is a real number*

1mn

mn

xx

3

43

4

3

116

16

1

21

8

34

34

314

3

116

161

16

1 1

2 8

*If 0x and n is even, then 1nx is not a real number.

Rational Exponents

, 0x

Write each expression in radical notation and evaluate.

14.3

216

15.2

38

Write each expression in radical notation and evaluate.

16. 2

38

Write each expression in radical notation and evaluate.

17. 3

481

Write each expression in radical notation and evaluate.

18.2

327

Write each expression in radical notation and evaluate.

19. 1

3125

Write each expression in radical notation and evaluate.

Objective 2: Use the properties of exponents.

You should be familiar with the properties of integer exponents from Chapter 5. Note that these properties apply to all rational exponents.Properties of Exponents

Let m and n be real numbers and x, xm, xn ,y, ym, and yn be nonzero real numbers.Product rule: m n m nx x x Product to a Power: m m mxy x y

Quotient rule: m

m nn

xx

x

Power rule: nm mnx x

Quotient to a Power: m m

m

x x

y y

Negative power:

n nx y

y x

Simplify each expression. Assume that x is a positive real number.

20. 7

6 275

Simplify each expression. Assume that x is a positive real number.

21.31

8 849 49

Simplify each expression. Assume that x is a positive real number.

22.1 1

3 34 2

Simplify each expression. Assume that x is a positive real number.

23.13

5

85

3

3

Simplify each expression. Assume that x is a positive real number.

24.3

8 416

81

x

Simplify each expression. Assume that x and y are positive real numbers.

25.

29 327

8

x

Simplify each expression. Assume that x and y are positive real numbers.

26. 9

1 2 29 99 9

Simplify each expression. Assume that x and y are positive real numbers.

27.

13

15

x

x

Simplify each expression. Assume that x and y are positive real numbers.

28.2 1

5 2x x

Simplify each expression. Assume that x and y are positive real numbers.

29.

62 7

3

53

x

x

Simplify each expression. Assume that x and y are positive real numbers.

30. 2

9 3 34 2125x y

Simplify each expression. Assume that x and y are positive real numbers.

31. 1 1

3 25 5

22 5

xy x y

x y

Simplify each expression.

32. 3 7 325 5 5 52 9 5x x x x

Simplify each expression.

33. 1 1 1 12 2 2 23 5 3 5x y x y

Simplify each expression.

34. 2123x x

Simplify each expression.

35. 1 2 13 3 33 3 9a a a

Use a graphing calculator or a spreadsheet to approximate each expression to the nearest hundredth.

36. 37. 38.1623

237

343