Rational Exponents

In other words, exponents that are

fractions.

Definition ofFor any real number b and any integer n > 1,

except when b < 0 and n is even

b1

n

b1

n = bn

Examples:

= 36 1. 361

2 =6

= 643 2. 64

1

3 =4

=

149

⎛

⎝ ⎜

⎞

⎠ ⎟

1

2

3. 49−

1

2

=

149

= 83

4.

18

⎛

⎝ ⎜

⎞

⎠ ⎟

−1

3

=2

Examples:

=

17

= 8( )

1

3

Economists refer to inflation as increases in the average cost of purchases. The formula C = c(1 + r)n can be used to predict the cost of consumer items at some projected

time. In this formula C represents the projected cost of the item at the given

annual inflation rate, c the present cost of the item and r is the rate of inflation (in decimal form), and n is the number of

years for the projection. Suppose a gallon of milk costs $2.69 now. How much would

the price increase in 6 months with an inflation rate of 5.3%?

Step 1: Identify the known values

Formula C = c(1 + r)n

c = $2.69 present cost of the item

r = 0.053 rate of inflation (in decimal

form)n = 1/2

# of years for the projection

Step 2: Find the value for C

Formula C = c(1 + r)n

C = 2.69(1 + 0.053)1/2

C = 2.69(1.053)1/2

C = $2.76

Answer the question How much would the price increase?

$2.76-$2.69 = $0.07 or 7¢

Definition of Rational Exponents

For any nonzero number b and any integers m and n

with n > 1,

except when b < 0 and n is even

b

m

n = bmn = bn( )

m

NOTE: There are 3 different ways to write a rational exponent

27

4

3 = 2743 = 273( )

4

2. 274

3

Examples:

= 273

( )4

1. 363

2

= 3( )

4 =81

= 36( )

3

= 6( )

3 =216

3. 813

4 = 814

( )3

= 3( )

3 =27

Simplifying Expressions No negative exponents No fractional exponents in the

denominator No complex fractions (fraction

within a fraction) The index of any remaining

radical is the least possible number

Examples: Simplify each expression

=42

6 ⋅a3

6 ⋅b5

6 41

3 ⋅a1

2 ⋅b5

6

= 426 ⋅ a36 ⋅ b56

= 16a3b56Get a common denominator - this is going to be our index

Rewrite as a radical

Examples: Simplify each expression

=x1

2+

3

4+

1

5

=x10

20+

15

20+

4

20 =x29

20

x1

2 ⋅x3

4 ⋅x1

5

=x20

20 ⋅x9

20 =x x920

Remember we add exponents

=w

1

5

w5

5

=w

1

5

w

Examples: Simplify each expression

w−4

5

=

1w

⎛

⎝ ⎜

⎞

⎠ ⎟

4

5

=1

4

5

w4

5

=1

w4

5

⋅w

1

5

w1

5

To rationalize the denominator

we want an integer exponent

=w

1

5

w4

5+

1

5 =

w5

w

=

xy7

8

y

Examples: Simplify each expression

=x

1y

⎛

⎝ ⎜

⎞

⎠ ⎟

1

8

=x⋅1

y1

8

=x

y1

8

⋅y

7

8

y7

8

To rationalize the denominator

we want an integer exponent

=x y78

y

xy−1

8

=21

2−

1

4 =22

4−

1

4 =21

4

Examples: Simplify each expression

=32

1

10

41

8

=25

( )1

10

22( )

1

8

=2

5

10

22

8

=2

1

2

21

4

3210

48

= 24

Examples: Simplify each expression

=5

−1

2

2⋅51

2

5−1

2

2 5

=12

⋅5

−1

2

51

2

=

12

⋅5−

1

2−

1

2

=

12

⋅5−1

=

12

⋅15

=110

Examples: Simplify each expression

⋅m

1

2 +1

m1

2 +1

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

=

m+1m−1

1

m1

2 −1

=

m1

2 +1m−1

Multiply by conjugate and use

FOIL

Examples: Simplify each expression

62

5 y8

5 =6

2

5 y5

5 y3

5 =6

2

5 ⋅ y⋅ y3

5

= y⋅ 625 ⋅ y35 = y 62 y35

= y 36y35

Examples: Simplify each expression

c2

3 −c

c−1

2

=c2

3 ⋅c1

2 −c⋅c1

2 =c7

6 −c3

2

=c

1

2 c2

3 −c⎛

⎝ ⎜

⎞

⎠ ⎟

Examples: Simplify each expression

=c6

6c1

6 −c2

2c1

2

=c c6 −c c

=c7

6 −c3

2

Examples: Simplify each expression

=2x

3

2

x2

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

−2

2x−2

x−3

2

⎛

⎝

⎜ ⎜

⎞

⎠

⎟ ⎟

−2

= 2x

3

2−2⎛

⎝ ⎜

⎞

⎠ ⎟

−2

= 2x

−1

2⎛

⎝ ⎜

⎞

⎠ ⎟

−2

=2−2x−1

2⋅−2

=

122 ⋅x

=

x4