7.4 Rational Exponents

I. Simplifying Expressions with Rational Exponents.

• Rational exponent – when there is a fraction as an exponent

• This is a different way of writing out a radical sign.

• How to interpret the fraction exponent:• Denominator – this is the index of the radical sign• Numerator – this is the power that the radicand is

raised to, or you can raise the whole radical expression to

a r/n = n√(ar) = (n√a)r

Fractional Exponents (Powers and Roots)

xyy xy

x

aaa )(“Power”

“Root”

Quick Review – Exponent Rules

NEGATIVE EXPONENTRULE

nn

a

1a

2525

1

Upstairs – Downstairs: A negative exponent means it’s in the wrong place.

PRODUCT OR POWERRULE

2010 22 nmnm aaa

302

HAVE TO HAVE THESAME BASE

QUOTIENT OF POWERRULE

4

10

3

3 63

bab

a

xx

x HAVE TO HAVE THESAME BASE

POWER OF POWERRULE

(x4)³ 34x 12x

mnnm a)a(

POWER OF PRODUCTRULE

mnnnm ba)ab(

(2x4)⁵ 545 x2 20x32

POWER OF QUOTIENT 2RULE

2

y

3

2

2

y

3

2

2

3

y

9

2y

a

aa

x

y

y

x

RATIONAL EXPONENTRULE

mnn mn

m

aaa

4

3

16 34 16 32 8

RADICAL TO EXPONENTRULE

251 2/

a a1 2/

25 5a an n1/

Let’s put a few ideas together

5

3

x

35

x 35

1

x

6.0x Convert the decimal to a fraction

EX - simplify

6

8

3

816

8 4

3

3

6 3

3 84

6 3

3 21

6 3

3

6 5

6 5

3

3

6 6

6 5

3

33

3

33 65

21

3

33 65

63

3

3 68

3

3 34

3

333 3 3

Need to Rationalize!

Remember: We are ADDING exponents here – what do we need to add fractions?

How did we get to here???

7.4 Real-World Connection - Example 3

• The optimal height h of the letters of a message printed on pavement is given by the formula

h = 0.00252d2.27

eHere d is the distance of the driver from the letters and e is the height of the driver’s eye above the pavement. All of the distances are in meters.

Find h for the given value of d and e.

d = 100 m; e = 1.2 m

Continued

h = 0.00252d2.27

e

H = 0.00252(100)2.27

1.2

Let’s Try Some

53

32

43

816

y

5.3

4

31

158

x

Hint: convert to a fraction rather than a decimal!

Let’s Try Some

53

32

43

816

y

5.3

4

31

158

x