Rational ExponentsRational Exponents

Fraction ExponentsFraction Exponents

Radical expression and Exponents

By definition of Radical Expression.

The index of the Radical is 3.

51251255 33 so

How would we simplify this expression?

What does the fraction exponent do to the number?

The number can be written as a Radical expression, with an index of the denominator.

2

1

9

2 9

The Rule for Rational Exponents

46464 33

1

1

nn bb

Write in Radical form

2

1

6

1

m

a

Write in Radical form

mm

aa

2

1

66

1

Write each Radical using Rational Exponents

w

b5

Write each Radical using Rational Exponents

2

1

5

15

ww

bb

What about Negative exponents

Negative exponents make inverses.

7

1

49

149

2

12

1

What if the numerator is not 1Always remember:

Evaluate

5 25

2

3232

power

root

What if the numerator is not 1

Evaluate

322

5 = 3225

25( )

25 = 2105

What if the numerator is not 1

Evaluate

322

5 3225

25( )

25 2105 2

105

22 4

power

root

For any nonzero real number b, and integer m and n

Make sure the Radical express is real, no b<0 when n is even.

bm

n = bmn or bn( )

m

Simplify each expression

No fraction in the denominators.

3

2

7

4

7

1

x

yy

Simplify each expression

No fraction in the denominators.

y1

7 ⋅y47 y

57 y57

x−23

1

x23

1

x23

Simplify

64x63

Simplify

64x63

(64x6 )1

3

641

3 (x6 )1

3 =4x63 =4x2

Simplify

54x93

Simplify in simplest radical form

54x93

27 • 2 • x93

27 • 2 • x9( )

13 =27

132

13 x

93

Radical form:

54x93

27 • 2 • x93

27 • 2 • x9( )

13 =27

132

13 x

93 =3x3 23

Simplifying exponential expressions

• If in radical form, transform to exponential form

• Simplify by applying the exponent to each base.

• Express with positive exponents or radical form, depending on the directions

Express in simplest form

• Make exponential first!

25x−4

36y8

Express in simplest form

• Make exponential first!

• Now simplify

25x−4

36y8=

25x−4

36y8

⎛

⎝⎜

⎞

⎠⎟

12=

2512 x

−42

3612 y

82

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

Express in simplest form

• Make exponential first!

• Now simplify

25x−4

36y8=

25x−4

36y8

⎛

⎝⎜

⎞

⎠⎟

12=

2512 x

−42

3612 y

82

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟

5x−2

6y4=

56x2y4

Next example

• Use positive exponents only!

x4y12

z−26

Next example

• Use positive exponents only!

• Or in radical form:

x4y12

z−26 =

x4y12

z−2

⎛

⎝⎜

⎞

⎠⎟

16=

x46 y

126

z−26

⎛

⎝

⎜⎜⎜

⎞

⎠

⎟⎟⎟=

x23y2

z−13

=x23y2z

13

y2 x2z3

Simplify

1

1

2

1

2

1

y

y

Simplify

1

1

1

1

2

1

2

1

2

1

2

1

y

y

y

y

Simplify

1

1

1

1

1

11

2

1

2

11

2

1

2

1

2

1

2

1

y

yyy

y

y

y

y

Simplify

1

12

1

1

1

1

1

1

2

1

1

2

1

2

11

2

1

2

1

2

1

2

1

y

yy

y

yyy

y

y

y

y