7.1 nth Roots and Rational Exponents

2/19/2014

nth RootEx. 32 = 9, then 3 is the square root of 9.

If b2 = a, then b is the square root of a.

If b3 = a, then b is the cube root of a.

If b4 = a, then b is the fourth root of a.

If bn = a, then b is the nth root of a.You can write the nth root of a as Where a is a real number and n is the index of the radical.

)93(

ab

3 ab

4 ab

Number of Real Roots(use for solving)

Examplen

Odd

Even

a

Any real number

Greater than 0

0

Less than 0

Number of Roots

One

Two

One

No Real Solution

283 3273

2 and 2164

002

solution no 92

𝑛√𝑎

Example 1 Find nth Root(s)

Find the indicated nth root(s) of a.

a. =n 3, =a 64– b.

=n 4, =a 81

SOLUTION

a. Because n is odd, 64 has one real cube root.–

b. Because n is even and a is greater than 0, 81 has two real fourth roots.

CHECK ( )34– = ( )4– ( )4– ( )4– = 64–

3 64–

4 81

𝑛√𝑎

¿− 4

¿3 𝑎𝑛𝑑−3

Extra Practice

Find the indicated nth root(s) of a.

1. =n 2, =a 144 ANSWER 12–12,

2. =n 3, =a 1000 ANSWER 10

3. =n 4, =a 256 ANSWER 4–4,

Example 2 Solve Equations Using nth Roots

Solve the equation.

a. =2x 4 162

SOLUTION

a. =2x 4 162 Write original equation.

=2 2

1622x 4Divide each side by 2.

=x 4 81

=x 3–+

4 =x 4 4 81 Take fourth root of each side.

Example 2 Solve Equations Using nth Roots

b.

Divide both sides by -2

= Cube root both sides

x = -5

When to have 1 answer instead of 2 answers when doing problems with

• When the problem says SOLVE, you may have 1 or 2 answers depending if the index is odd or even.

• When the problem says EVALUATE, then you only have 1 answer.

http://www.khanacademy.org/math/algebra/exponent-equations/fractional-exponents-tut/v/basic-fractional-exponents

Ex :

Radical Form:

VocabularyRational Exponents: exponents written as fractions

2

1

a 3

1

a 4

1

a

aa 2 3 a 4 a

In general:nn aa

1

Example 3

Evaluate the expression.

Evaluate Expressions with Rational Exponents

a. 91/2 = 9 = 3

b. 161/4 = 2= 164

c. 641/3 = 4= 643

( )1/432–d. = 324 – , no real solution

Extra Practice

Evaluate the expression.

ANSWER 5

Evaluate Expressions

4. 251/2

5. 811/2 ANSWER 9

6. 1251/3 ANSWER 5

7. 321/5 ANSWER 2

http://www.khanacademy.org/math/algebra/exponent-equations/fractional-exponents-tut/v/fractional-exponents-with-numerators-other-than-1

Ex :

Radical Form:

Rational Exponents(when numerator is not 1)

2

3

a

32 a 54 a

4

5

a

In general: nmm

n

aa

Note: denominator is the index of the radical and numerator is the exponent of the radical

3

2

1

a

5

4

1

a

Ex :

Radical Form:

Negative Rational Exponentnegative exponent still “moves” power

2

3

a

32

1

a 54

1

a

4

5

a

In general:

nm

m

n

aa

1

2

3

1

a

4

5

1

a

Example 4 Rewrite Expressions

( )4 35 = 53/4

Rewrite using rational exponents.( )4 35a.

b. Rewrite using radicals.72/5

72/5 = ( )5 27

=1

( )3 22

Rewrite using radicals. 2 2/3–

2 2/3– =1

22/3

c.

Extra Practice

Rewrite the expression using rational exponents.

8. ( )5 22 ANSWER 22/5

9.1

134ANSWER 13 1/4–

Extra Practice

Rewrite the expression using radicals.

10. 152/3

11. 11 1/3– ANSWER1

113

ANSWER 15( )3 2

12. 29 2/5– ANSWER1

29( )5 2

Example 5 Evaluate Expressions with Rational Exponents

Evaluate the expression.

a. 43/2 b. 8 2/3–

SOLUTION

Use radicals to rewrite and evaluate each expression.

a. 43/2 3= ( )4 = 23 8=

b. 8 2/3– =1

82/3=

1

( )3 28=

1

22=

4

1

Checkpoint

Evaluate the expression.

17. 253/2

18. 165/4

19. 8 5/3–

ANSWER 125

ANSWER 32

ANSWER1

32

Rewrite and Evaluate Expressions with Rational Exponents

Homework: WS 7.1

Odd problems only, skip #11

“I tried to catch some fog. I mist!”