3.1 Evaluate nth Roots and Use Rational Exponents

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3.1 Evaluate nth Roots and Use Rational Exponents. p. 166 What is a quick way to tell what kind of real roots you have? How do you write a radical in exponent form? What buttons do you use on a calculator to approximate a radical? What is the difference between evaluating and solving?.

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### 3.1 Evaluate nth Roots and Use Rational Exponents

p. 166

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

How do you write a radical in exponent form?

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

What is the difference between evaluating and solving?

Real nth Roots

Let n be an integer greater than 1 and a be a real number.

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 one nth root.

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

See page 166 for KEY CONCEPT

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.

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

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

a. n = 3, a = –216

b. n = 4, a = 81

SOLUTION

Find the indicated real nth root
• n = 3, a = −125
• n = 4, a = 16
Rational Exponents

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

See page 167 for KEY CONCEPT

1

1

23

323/5

64

( )3

=

(161/2)3

=

43

=

43

=

64

=

=

16

1

1

1

1

1

1

=

=

=

=

(321/5)3

323/5

( )3

5

8

23

8

32

=

=

=

=

Evaluate (a) 163/2 and (b)32–3/5.

SOLUTION

Rational Exponent Form

a. 163/2

163/2

b. 32–3/5

32–3/5

Keystrokes

Expression

Display

7 3 4

9 1 5

12 3 8

7

c. ( 4 )3 = 73/4

Approximate roots with a calculator

a. 91/5

1.551845574

1

5

b. 123/8

2.539176951

3

8

4.303517071

4

3

Using a calculator to approximate a root

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

Keystrokes

Expression

Display

4 2 5

1

64

- 2 3

10. 64 2/3

11. (4√ 16)5

–30 2 3

12. (3√–30)2

16 5 4

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

9. 42/5

1.74

0.06

32

9.65

Solve the equation using nth roots.
• 2x4 = 162

x4 = 81

x4 = 34

x = ±3

• (x − 2)3 = 10

x ≈ 4.15

x5 = 512

1

1

x5

= 512

2

2

x5

= 1024

x

= 5 1024

x

= 4

SOLUTION

Multiply each side by 2.

take 5th root of each side.

Simplify.

( x – 2 )3

= –14

( x – 2 )

= 3 –14

x

= 3 –14 + 2

x

= 3 –14 + 2

x

= – 0.41

( x – 2 )3 = –14

SOLUTION

Use a calculator.

( x + 5 )4

= 16

( x + 5 )

= +4 16

x

= + 4 16 – 5

or

x

= 2 – 5

x

= – 2 – 5

x

or

x

= – 3

= –7

( x + 5 )4 = 16

SOLUTION

take 4th root of each side.

Write solutions separately.

Use a calculator.

Evaluating a model with roots.

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

Solving an equation using an nth root.

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

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

• How do you write a radical in exponent form?

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

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

Root buttons

• What is the difference between evaluating and solving?

Evaluating simplifies; Solving finds answers x=.

Assignment
• Page 169, 9-45 every 3rd problem, 50-56 even,

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