OBJECTIVE:The students will simplify

expressions by using the laws of exponents.

Write in exponential form

3x32x2x2x2x2x210x10x105x5x51x1x1x1x1x1x1x1

HISTORY of the word HISTORY of the word EXPONENT.EXPONENT.

The term EXPONENT The term EXPONENT was introduced by was introduced by Michael Stifel (1487-Michael Stifel (1487-1567) in 1544 in 1567) in 1544 in Arithmetica integra. Arithmetica integra.

An exponent is simply shorthand for multiplying that number of identical factors.

So 4³ is the same as (4)(4)(4), three identical factors of 4. And x³ is just three factors of x, (x)(x)(x).

Exponent is the power in Exponent is the power in an expression.an expression.

1313 77

ExponentsExponents

35power

base

exponent

3 3 means that is the exponential

form of t

Example:

he number

125 5 5

.125

Using the CalculatorUsing the Calculator

5 5 44

Press 5Press 5

Press Press ^̂Press 4Press 4

Then =Then =

7 Laws of Exponents #17 Laws of Exponents #1PRODUCT LAWPRODUCT LAW

To To MultiplyMultiply LIKE Bases… LIKE Bases…

……Copy the Base, Add Exponents Copy the Base, Add Exponents

Product Law or Product RuleProduct Law or Product Rule

m n m nx x x 3 4 3 4 7Example: 2 2 2 2

3 4

7

Proof: 2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2

7 Laws of Exponents #27 Laws of Exponents #2QUOTIENT LAWQUOTIENT LAW

To Divide LIKE Bases…To Divide LIKE Bases…

……Subtract ExponentsSubtract Exponents

6

2

a

a a4

Quotient Law or Quotient Rule:Quotient Law or Quotient Rule:

mm n m n

n

xx x x

x

44 3 4 3 1

3

5Example: 5 5 5 5 5

5

4

3

5 5 5 5 5Proof: 5

5 5 5 5

7 Laws of Exponents #37 Laws of Exponents #3EXPONENT of EXPONENT LAWEXPONENT of EXPONENT LAW

To Raise a Power to a Power…To Raise a Power to a Power…

……Multiply ExponentsMultiply Exponents

43a a12

Exponent of Exponent Law or Exponential Rule:Exponent of Exponent Law or Exponential Rule:

nm mnx x

23 3 2 6Example: 4 4 4

2 23

6

Proof: 4 4 4 4 4 4 4 4 4 4

4 4 4 4 4 4 4

To Raise a QUANTITY To Raise a QUANTITY to a Power, raise EACH to a Power, raise EACH Factor to that Power.Factor to that Power.

(-3ab)2= 9a2b2

7 Laws of Exponents #47 Laws of Exponents #4Raising a product to a Raising a product to a

powerpower

7 Laws of Exponents #57 Laws of Exponents #5Raising a quotient or a Raising a quotient or a

fraction to a powerfraction to a power To Raise a FRACTION To Raise a FRACTION

to a Power, raise BOTH to a Power, raise BOTH Numerator & Denominator Numerator & Denominator to that power. to that power.

4a

b

a4

b4

n n

n

x x

y y

3 3

3

2 2Example:

7 7

7 Laws of Exponents #67 Laws of Exponents #6NEGATIVE EXPONENT LAWNEGATIVE EXPONENT LAW

Negative Exponents Negative Exponents

……Reciprocal with a Positive ExponentReciprocal with a Positive Exponent

3a a3

#6: Negative Law of Exponents: If the base is powered by the negative exponent, then the base becomes reciprocal with the positive exponent.

1mm

xx

33

1 1Example #1: 2

2 8

33

3

1 5Example #2: 5 125

5 1

7 Laws of Exponents #77 Laws of Exponents #7

Any nonzero number raised to the ZERO Power = Any nonzero number raised to the ZERO Power = ONEONE

0a

02

03,263,546

The Laws of Exponents:The Laws of Exponents:

#7: Zero Law of Exponents: Any base powered by zero exponent equals one

0 1x

0

0

0

Example: 112 1

51

7

1flower

3

y

x3

3

y

x

4

7

x

x

1

47x 3x

7

5

x

x 57

1

x 2

1

x

Basic Examples

Basic Examples

32 xx 32x 5x

34x 34x 12x

3xy 33yx