Properties of Exponents

Zero Power Property a0 = 1

Product of Powers Property am • an = am+n

Power of Power Property (am)n = am•n

Negative Power Property a-n = 1/an, a 0

Power of Product Property (ab)m = ambm

Quotients of Powers Property

am

an am n , a 0

Power of Quotient Property

(a

b)m

am

bm , b 0

Rational ExponentsMr. Joyner

Tidewater Community College

Intermediate Algebra -MTH04

Intermediate Algebra MTH04

Radicals (also called roots) are directly related to exponents.

Rational Exponents

Intermediate Algebra MTH04

All radicals (roots) can be written in a different format without a radical symbol.

Rational Exponents

7.1 – RadicalsRadical Expressions

Finding a root of a number is the inverse operation of raising a number to a power.

This symbol is the radical or the radical sign

n a

index radical sign

radicand

The expression under the radical sign is the radicand.

The index defines the root to be taken.

Intermediate Algebra MTH04

This different format uses a rational (fractional) exponent.

Rational Exponents

Intermediate Algebra MTH04

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 aa1

Intermediate Algebra MTH04

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

13 1111

base

Intermediate Algebra MTH04

Rational Exponents

For any exponent of the radicand, the rational exponent form of a radical looks like this:

n

mm

nn m aaa

How do you simplify ?

Intermediate Algebra MTH04

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

Intermediate Algebra MTH04

Rational Exponents

Example:

2

1

16 416

3

1

125 51253

Intermediate Algebra MTH04

Rational Exponents

Examples:

5

2

32 4232 225

3

5

64 1024464 553

Intermediate Algebra MTH04

Rational Exponents

Examples:

No real number solution 2

1

16 16

3

2

216 366216 223

Rational Exponents

More Examples:

32

32

27

132

27

1

3 2

3 2

27

19

13

3

729

1

32

32

27

132

27

1

23

23

27

1

9

1 2

2

3

1

or

Intermediate Algebra MTH04

Rational Exponents

The basic properties for integer exponents also hold for rational exponents as long as the expression represents a real number.

See the chart on page 389 of your text.

Intermediate Algebra MTH04

Rational Exponents

Example:

What would the answer above be if you were to write it in radical form?

6

16

3

6

4

2

1

3

2

2

1

3

2

555

5

5

Intermediate Algebra MTH04

Rational Exponents

Example:

6

16

3

6

4

2

1

3

2

2

1

3

2

555

5

5

6 5

Intermediate Algebra MTH04

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.

Intermediate Algebra MTH04

The Square Root Rules (Properties)

Rational Exponents

b

a

b

ababa

Multiplication Division

b may not be equal to 0.

Intermediate Algebra MTH04

The Cube Root Rules (Properties)

Rational Exponents

33

3

b

a

b

a333 baba

Intermediate Algebra MTH04

Rational Exponents

The more general rules for any radical are as follows …

Multiplication Division

b may not be equal to 0.

Intermediate Algebra MTH04

The Rules (Properties)

Rational Exponents

nn

n

b

a

b

annn baba

Intermediate Algebra MTH04

Rational Exponents

These same rules in rational exponent form are as follows …

Multiplication Division

b may not be equal to 0.

Intermediate Algebra MTH04

The Rules (Properties)

Rational Exponents

nnn baba111

n

n

n

b

a

b

a1

1

1

Intermediate Algebra MTH04

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?

Intermediate Algebra MTH04

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.

Intermediate Algebra MTH04

Rational Exponents

Examples – Simplifying Radical Expressions:

3 54 3 227 33 227 3 23

5

3

3 6

8

5

5

5

3

25

155

15

3 2

3 2

3 6

6

6

8

3 3

3 2

6

68 6

68 3 2

3

64 3 2