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Algebra 2: Section 7.1

Nth Roots and Rational Exponents

7.1: Evaluating nth Roots• You are familiar with square roots

– 2 is the square root of 4, since 22 = 4.

• This concept can be extended to other types of roots– 2 is the cube root of 8, since 23 = 8– 2 is the fourth root of 16, since 24 = 16– 2 is the fifth root of 32, since 25 = 32– 2 is the nth root of a, since 2n = a

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Rational Exponents1

nna am

nam

na

1 m

na

( )mn a

1m

na

1

( )mn a

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Examples

• Rewrite using rational exponents.

• Rewrite using radical notation.

5

3. 75

27

284. 3

2

831

43

4

35. 7 43 7

31. 271

327

52. a1

5a

4

56. 4 45 4

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Nth Roots Rules for Signs of Answers

• For odd roots (3, 5, 7, etc)

– There is ALWAYS ONLY ONE answer (either + or -)

• For even roots (2, 4, 6, etc)

– If the radical symbol is already in the problem, then use the sign that is in front

– If YOU put the radical symbol in the problem to solve an equation, then there will be two answers (+ and -)

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Examples

• Find the nth root(s) of a7. n = 5, a = -32

8. n = 3: a = 64

9. n = 4: a = 256

10. n = 2: a = 169

What multiplied by itself 5 times equals -32?

What multiplied by itself 3 times equals 64?

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4

±13

What multiplied by itself 4 times equals 256?

What multiplied by itself 2 times equals 169?

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ExamplesSimplify or evaluate each expression:

7

311. 125 5

2412. 16

3613. 0

22 4

30 0

Rational Exponents (examples)

• Simplify or evaluate each expression:

14. 82/3

15. 163/4

16. 4-5/2

3 64 4

34 16 32 8

52

1

4

5

1

2

1

32

23 8

9

Examples

• Solve the equation. Round your answer to two decimal places.

3 8x

2x

317. 5x = 40

Examples

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35 27x

5 3x

2x

318. ( 1) 32y 31 32y

319. 3(x +5) = 81

4.17

3 32 1y

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Homework

P.404

#13-22 all

#24-40 evens

#53-61 all

List of PerfectsPerfect Squares:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225

Perfect Cubes:

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000

Perfect Quads:

1, 16, 81, 256, 625

Perfect Fifths:

1, 32, 243, 1024, 3125