• O B J E C T I V E S : S T U D E N T S W I L L B E A B L E T O …

• E V A L U A T E N T H R O O T S O F R E A L N U M B E R S U S I N G R A D I C A L N O T A T I O N A N D R A T I O N A L E X P O N E N T N O T A T I O N .

7.1: NTH ROOTS AND RATIONAL EXPONENTS

nth ROOT OF a

• If bn = a, then b is the nth root of a• 33 = 27, 3 is the cube root of 21• 24 = 16, 2 is the 4th root of 16

RULE:

n is called the “index”. It tells you what root you are taking.

EX:

nn aa1

3

13 88

REAL NTH ROOTS

Let n be an integer > 1, and let a be a real number:• If n is odd, then a has 1 nth real root:

• If n is even, and a > 0, then a has 2 real nth roots:

• If n is even and a = 0, then a has 1 nth root

• If n is even and a < 0, then a has no real nth roots no real roots

2)8(8 3

13

21616 4

14

000 2

1

4

FIND THE INDICATED NTH ROOT(S)

64,6.)4

81,4.)3

64,3.)2

32,5.)1

an

an

an

an

RATIONAL EXPONENTS

• Can rewrite a radical expression with rational exponents. They are one in the same!

Example: OR

mnn mn

m

aaa

27399 332

3

2772999 32

3

USING NEGATIVE EXPONENTS

n mmn

n

mn

m

aaa

a111

9

1

27

1

27

1

27

127

3 223

3

23

2

EXAMPLES

Rewrite using radical notation:

Rewrite using rational exponents:

3 211

5

2

5

EVALUATE:

4

3

2

3

3

2

2

5

16.)4

49.)3

64.)2

16.)1

USING YOUR CALCULATOR TO EVALUATE

4

3

3 Enter 3 ^ (3 / 4)

SOLVING EQUATIONS USING NTH ROOTS

Isolate the quantity raised to the nth power, and take the nth root of both sides.

Be careful….If n is odd, 1 solutionIn n is even, 2 solutions (±)

EXAMPLES: SOLVE

1025.)3

321.)2

805.)1

8

3

4

x

y

y

EXAMPLE

A basketball has a volume of about 455.6 in3. The formula for the volume of a basketball is V = 4.18879r3. Find the radius of the basketball.

EXAMPLE

The rate r at which an initial deposit P will grow to a balance A in t years with interest compounded n times a year is given by

Find r if P = $1000, A = $2000, t= 11 years, and n = 12

1

1

nt

P

Anr