Rational Exponents

AAT-A

Date: 12/11/13 SWBAT write and simplify rational exponents in

radical form and vice versa.

Do Now: pg 254 #43-48

HW Requests:

Continue Vocab sheet

In class: pg 261 #4-19

HW: Complete Division of Radicals WS #1-16; pg 261 #21-41 odds

Continue Vocab sheet

In class: pg 261 #4-19

Complete Announcements:

Math Team Cancelled Wed.

Tutoring: Tues. and Thurs. 3-4

"Do not judge me by my successes,

judge me by how many times

I fell down and got back up again.“

Nelson Mandela

Radicals (also called roots) are

directly related to exponents.

Rational Exponents

All radicals (roots) can be

written in a different format

without a radical symbol.

Rational Exponents

This different format uses a

rational (fractional) exponent.

When the exponent of the radicand

(expression under the radical symbol)

is one, the rational exponent form of a

radical looks like this:

Rational Exponents

Remember that the index, n, is a whole number

equal to or greater than 2.

nn aa

1

Rational Exponents

Examples:

• When a base has a fractional exponent, do not

think of the exponent in the same way as when it is

a whole number.

• When a base has a fractional exponent, the

exponent is telling you that you have a radical

written in a different form.

2

1

66 3

1

3 1111base

Rational Exponents

For any exponent of the radicand, the

rational exponent form of a radical

looks like this:

n

mm

nn m aaa Power

Root or index

How do you simplify ?

Rational Exponents

• You can rewrite the expression using a radical.

• Simplify the radical expression, if possible.

• Write your answer in simplest form.

• Reduce the rational exponent, if possible.

2

1

16

Rational Exponents

Example:

2

1

16 416

3

1

125 51253

Rational Exponents

Examples:

5

2

32 423222

5

3

5

64 102446455

3

Rational Exponents

Examples:

No real number solution 2

1

16 16

3

2

216 36621622

3

Rational Exponents

The basic properties for integer

exponents also hold for rational

exponents as long as the expression

represents a real number.

Rational Exponents

Example:

What would the answer above be if you

were to write it in radical form? Cont.

6

1

6

3

6

4

2

1

3

2

2

1

3

2

555

5

5

Rational Exponents

Example:

6

1

6

3

6

4

2

1

3

2

2

1

3

2

555

5

5

6 5

Rational Exponents

Simplifying radicals – A radical expression is in

simplest form once ALL of the following

conditions have been met.…

• the radicand (expression under the radical symbol) cannot

be written in an exponent form with any factor having an

exponent equal to or larger than the index of the radical;

• there is no fraction under the radical symbol;

• there is no radical in a denominator.

Rational Exponents

Do you remember the basic Rules of

Exponents that you learned in Roots

and Radicals?

See the next two slides for a quick

review.

Multiplication Division

b may not be equal to 0.

The Square Root Rules (Properties)

Rational Exponents

b

a

b

ababa

Multiplication Division

b may not be equal to 0.

The Cube Root Rules (Properties)

Rational Exponents

33

3

b

a

b

a

333 baba

Rational Exponents

The more general rules for any

radical are as follows …

Multiplication Division

b may not be equal to 0.

The Rules (Properties)

Rational Exponents

nn

n

b

a

b

a

nnn baba

Rational Exponents

These same rules in rational exponent

form are as follows …

Multiplication Division

b may not be equal to 0.

The Rules (Properties)

Rational Exponents

nnn baba111

n

n

n

b

a

b

a1

1

1

Rational Exponents

In working with radicals, whether in radical

form or in fractional exponent form, simplify

wherever and whenever possible.

What is the process for simplifying radical

expressions?

Rational Exponents

Examples – Simplifying Radical Expressions:

3 54 3 227 33 227 3 23

5

3

3 6

8

5

5

5

3

25

15

5

15

3 2

3 2

36

6

6

8

3 3

3 2

6

68

6

68 3 2

3

64 3 2

This different format uses a

rational (fractional) exponent.

Rational Exponents