Exponent Properties: Product, Quotient, Power, and Negative Exponents Rules, Slides of Algebra

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§1.6 Exponent

Properties

Review §

 Any QUESTIONS About

• §1.5 → (Word) Problem Solving

 Any QUESTIONS About HomeWork

• §1.5 → HW-

1.5 MTH 55

Quick Test of Product Rule

m n m n

a a a

⋅ =

 Test 2 3 5

? 2 3 3 ⋅ 3 = 3 = 3



35 = 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 = ( 3 ⋅ 3 ) (⋅ 3 ⋅ 3 ⋅ 3 ) = 9 ⋅ 27 = 243

Example  Product Rule

• Multiply and simplify each of the following. (Here “simplify” means express the product as one base to a power whenever possible.)

a) x^3 ⋅ x^5 b) 6 2 ⋅ 67 ⋅ 63

c) ( x + y )^6 ( x + y )^9 d) ( w^3 z^4 )( w^3 z^7 )

Exponent QUOTIENT Rule

• For any nonzero number a and any positive integers m & n for which m > n , m n n

m

a a

a (^) −

 In other Words: To DIVIDE powers with the same base, SUBTRACT the exponent of the denominator from the exponent of the numerator

Quick Test of Quotient Rule

 Test 6 4 2

? 4

6 5 5 5

5 = =

−



m n n

m

a a

a (^) −

5 5 5 5

5 5 5 5 5 5

5

5 4

6

⋅ ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅ = = 5 ⋅ 5 = 52 = 56 −^4

Example  Quotient Rule

• Solution a)

(^99 ) 3

x (^) x x

= − = x^6

 Solution b)

(^77 ) 3

(^8 ) 8

= − =^84

 Solution c)

(^1414 6 ) 6

(6 ) (^) (6 ) (6 ) (6 )

y (^) y y y

= − =

 Solution d)

7 9 7 9 3 3

6 6 4 4

r t r t r t r t

= ⋅ ⋅ 7 3 9 1 4 8 4

6 3 2

= ⋅ r −^ ⋅ t − = r t

Base is x

Base is 8

Base is (6 y )

TWO Bases: r & t

The Exponent Zero

• For any number a where a ≠ 0 1

0 a =

 In other Words: Any nonzero number raised to the 0 power is 1

• Remember the base can be ANY Number - 0.00073, 19.19, −86, 1000000, anything

The POWER Rule

• For any number a and any whole numbers m and n

m n m n

a a

 In other Words: To RAISE a POWER to a POWER, MULTIPLY the exponents and leave the base unchanged

Quick Test of Power Rule

 Test ( ) 2 3 6

3? 2 7 = 7 = 7

⋅

( 7 ) ( 49 )^3 ( 49 ) ( 49 ) ( 49 )

2 3 = = ⋅ ⋅

m n m n

a a

= ( 7 ⋅ 7 ) (⋅ 7 ⋅ 7 ) (⋅ 7 ⋅ 7 ) = 7 ⋅ 7 ⋅ 7 ⋅ 7 ⋅ 7 ⋅ 7 = 76

= 7 2 ⋅^3

Raising a Product to a Power

• For any numbers a and b and any whole number n ,

( )

n n n

a ⋅ b = a ⋅ b

 In other Words: To RAISE A PRODUCT to a POWER, RAISE Each Factor to that POWER

Quick Test of Product to Power

 Test ( ) 3 3

3? 2 ⋅ 11 = 2 ⋅ 11

( 2 ⋅ 11 )^3 = ( 22 )^3 = ( 22 ) (⋅ 22 ) (⋅ 22 ) = 10648

 2 11 8 1331 10648

( )

n n n

a ⋅ b = a ⋅ b

Raising a Quotient to a Power

• For any real numbers a and b , b ≠ 0, and any whole number n

n

n n

b

a

b

a  = 

  



 In other Words: To Raise a Quotient to a power, raise BOTH the numerator & denominator to the power

Quick Test of Quotient to Power

 Test

3

(^3)? 3

7

5

7

5  = 

  



3

(^33)

7



n

n n

b

a

b

a  = 

  

 3

(^33)

7

5

7

5  = 

  

