Algebra 2 Bellwork – 3/4/15

6.1 Evaluate nth Roots and Use Rational Exponents

Learning Target:

• Evaluate nth roots of real numbers using both radical notation and rational exponent notation.

The nth Root

n a

Index Number

Radicand

Radical

1na

The index number becomes the denominator of the exponent.

n > 1

Radicals

• If n is odd – one real root.• If n is even and

a > 0 Two real roots a = 0 One real root a < 0 No real roots

n a

Radical form to Exponential Form

23 x

Change to exponential form.

23x

or

213x

or

1

2 3x

Exponential to Radical Form

23x

Change to radical form.

223 3 or xx

The denominator of the exponent becomes the index number of the radical.

Ex. 1 Finding nth Roots

• Find the indicated real nth root(s) of a. A. n = 3, a = -125

Solution: Because n = 3 is odd, a = -125 has one real cube root. Because (-5)3 = -125, you can write:

51253 or 5)125( 31

Ex. 2 Finding nth Roots

• Find the indicated real nth root(s) of a. A. n = 6, a = 729

Solution: Because n = 6 is even, a = 729 has two real sixth roots. Because 36 = 729,

you can write:

37296 or

Example 3: Evaluate Without a Calculator

Evaluate without a calculator.

531. 8 5

33 2

52

32

42. 32 54 2

44 2 2

42 2

Ex. 4 Approximating a Root with a Calculator

• Use a scientific calculator to approximate:

34 )5(

SOLUTION: First rewrite as . Then enter the following:

43

5

To solve simple equations involving xn, isolate the power and then take the nth root of each side.

34 )5(

Ex. 5 Evaluating Expressions with Rational Exponents

A.

B.

273)9(9 3323

273)9(9 3321

23

Using radical notation

Using rational exponent notation.OR

4

1

2

1

)32(

1

32

132

22552

52

4

1

2

1

)32(

132

2251

52

OR

Example 6: Solving an equation

Solve the equation:

4 7 9993x 4

4

7 7 9993 7

10000

x

x

44 4 10000

10

x

x

Note: index number is even, therefore, two answers.

Ex. 7 & 8 Solving Equations Using nth Roots

A. 2x4 = 162B. (x – 2)3 = 10

3

81

81

1622

4

4

4

x

x

x

x

15.4

210

102-x

10 2)– (x

3

3

3

x

x

Homework due 3/4/15

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