3.1 Evaluate nth Roots and 3.1 Evaluate nth Roots and Use Rational ExponentsUse Rational Exponents

p. 166p. 166

What is a quick way to tell what kind of real What is a quick way to tell what kind of real roots you have?roots you have?

How do you write a radical in exponent form?How do you write a radical in exponent form?

What buttons do you use on a calculator to What buttons do you use on a calculator to approximate a radical?approximate a radical?

What is the difference between evaluating and What is the difference between evaluating and solving?solving?

Real nth RootsReal nth RootsLet n be an integer greater than 1 and a be a real Let n be an integer greater than 1 and a be a real

number.number.

If n is odd, then a has one real nth root.If n is odd, then a has one real nth root.

If n is even and a > 0, then a has two real nth roots.If n is even and a > 0, then a has two real nth roots.

If n is even and a = 0, then a has one nth root.If n is even and a = 0, then a has one nth root.

If n is even and a < o, then a has no real nth roots.If n is even and a < o, then a has no real nth roots.

See page 166 for KEY CONCEPTSee page 166 for KEY CONCEPT

nn aa 1

nn aa 1

000 1 nn

Find the indicated real nth root(s) of a.

a. n = 3, a = –216 b. n = 4, a = 81

SOLUTION

b. Because n = 4 is even and a = 81 > 0, 81 has two real fourth roots. Because 34 = 81 and (–3)4 = 81, you can write ±4√ 81 = ±3

a. Because n = 3 is odd and a = –216 < 0, –216 has one real cube root. Because (–6)3 = –216, you

can write = 3√ –216 = –6 or (–216)1/3 = –6.

Find the indicated real nth rootFind the indicated real nth root

n = 3, a = n = 3, a = −125−125

n = 4, a = 16n = 4, a = 16

55125 3 33

2216 4 44

Rational ExponentsRational Exponents

Let aLet a1/n1/n be an nth root of a, and let m be a be an nth root of a, and let m be a positive integer.positive integer.

mnmnnm aaa 1

0,111

1 a

aaaa m

nmnnm

nm

See page 167 for KEY CONCEPT

Evaluate (a) 163/2 and (b) 32–3/5.

SOLUTION

Rational Exponent Form Radical Form

a. 163/2 (161/2)3 = 43= 64= 163/2 ( )3= 16 43= 64=

b. 32–3/5 = 1323/5 = 1

(321/5)3

= 123

18

=

32–3/5 1323/5= 1

( )35 32=

= 123

18

=

Evaluate the expression with Evaluate the expression with Rational ExponentsRational Exponents

993/23/2

3232-2/5-2/5

2739 33

4

1

2

1

2

1

32

1

32

122

5 52

552

Approximate roots with a Approximate roots with a calculatorcalculator

Expression Keystrokes Display

a. 91/5 9 1 5 1.551845574

b. 123/8 12 3 8 2.539176951

7c. ( 4 )3 = 73/4 7 3 4 4.303517071

1 5

83

3 4

Using a calculator to Using a calculator to approximate a rootapproximate a root

34 5

Rewrite the problem as 53/4 and enter using ^ or yx key for the exponent.

34.3

Expression Keystrokes Display

9. 42/5 4 2 5 1.74

- 2 31

640.06

16 5 4 32

–30 2 3 9.65

10. 64 2/3–

11. (4√ 16)5

12. (3√ –30)2

Evaluate the expression using a calculator. Round the result to two decimal places when appropriate.

Solve the equation using nth roots.Solve the equation using nth roots.

2x2x44 = 162 = 162

xx44 = 81 = 81

xx44 = 34 = 34

x = x = ±3±3

(x (x − 2)− 2)33 = 10 = 10 3 102x2103 x

x ≈ 4.15

12

x5 = 512

SOLUTION

12

x5 = 512

Multiply each side by 2.x5 = 1024

take 5th root of each side. x = 5 1024

Simplify.x = 4

( x – 2 )3 = –14

SOLUTION

( x – 2 )3= –14

( x – 2 ) = 3 –14

x = 3 –14 + 2

x = 3 –14 + 2

x = – 0.41 Use a calculator.

( x + 5 )4 = 16

SOLUTION

( x + 5 )4 = 16

take 4th root of each side. ( x + 5 ) = + 4 16

add 5 to each side. x = + 4 16 – 5

Write solutions separately. x = 2 – 5 or x = – 2 – 5

Use a calculator.x = – 3 or x = –7

Evaluating a model with roots.Evaluating a model with roots.

When you take a number to with a rational When you take a number to with a rational exponent and express it in an integer exponent and express it in an integer answer, you have evaluated.answer, you have evaluated.

Solving an equation using an nth root.Solving an equation using an nth root.

When you have an equation with value that When you have an equation with value that has a rational exponent, you solve the has a rational exponent, you solve the equation to find the value of the variable.equation to find the value of the variable.

What is a quick way to tell what kind of real roots What is a quick way to tell what kind of real roots you have?you have?

Root is odd, 1 answer; root is even, 1 or 2 real Root is odd, 1 answer; root is even, 1 or 2 real answers.answers.

How do you write a radical in exponent form?How do you write a radical in exponent form?

Use a fraction exponent (powers go up, roots go Use a fraction exponent (powers go up, roots go down)down)

What buttons do you use on a calculator to What buttons do you use on a calculator to approximate a radical?approximate a radical?

Root buttonsRoot buttons

What is the difference between evaluating and What is the difference between evaluating and solving?solving?

Evaluating simplifies; Solving finds answers x=.Evaluating simplifies; Solving finds answers x=.

AssignmentAssignment

Page 169, 9-45 every 3Page 169, 9-45 every 3rdrd problem, 50-56 problem, 50-56 even, even,

To get credit for doing the problem, you To get credit for doing the problem, you must show the original problem along with must show the original problem along with your answer unless it is a calculator your answer unless it is a calculator problem (41-51)problem (41-51)