Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

[email protected] • MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§1.6 §1.6 ExponentExponentPropertiesProperties



Review §Review §

Any QUESTIONS About• §1.5 → (Word) Problem Solving

Any QUESTIONS About HomeWork• §1.5 → HW-01

1.5 MTH 55



Exponent PRODUCT RuleExponent PRODUCT Rule

For any number a and any positive integers m and n,

nmnm aaa In other Words:

To MULTIPLY powers with the same base, keep the base and ADD the exponents

Exponent

Base



Quick Test of Product RuleQuick Test of Product Rule

nmnm aaa Test 532

?32 3333

24327933 32

243279333333333335



Example Example Product Rule Product Rule

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

a) x3 x5 b) 62 67 63

c) (x + y)6(x + y)9 d) (w3z4)(w3z7)



Example Example Product Rule Product Rule

Solution a) x3 x5 = x3+5 Adding exponents

= x8

Solution b) 62 67 63 = 62+7+3

= 612

Solution c) (x + y)6(x + y)9 = (x + y)6+9

= (x + y)15

Solution d) (w3z4)(w3z7) = w3z4w3z7

= w3w3z4z7

= w6z11

Base is x

Base is 6

Base is (x + y)

TWO Bases: w & z



Exponent QUOTIENT RuleExponent QUOTIENT Rule

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

n

m

aa

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 RuleQuick Test of Quotient Rule

Test246

?

4

6

555

5

nmn

m

aa

a

5555

555555

5

54

6

462 5555



Example Example Quotient Rule Quotient Rule

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

• a) b)

• c) d)

9

3

x

x

7

3

8

8

14

6

(6 )

(6 )

y

y

7 9

3

6

4

r t

r t



Example Example Quotient Rule Quotient Rule

Solution a)9

9 33

xx

x 6x

Solution b)7

7 33

88

8 48

Solution c)14

14 6 86

(6 )(6 ) (6 )

(6 )

yy y

y

Solution d)7 9 7 9

3 3

66

4 4

r t r t

tr t r

7 3 9 41 8

4

36

2r t r t

Base is x

Base is 8

Base is (6y)

TWO Bases: r & t



The Exponent ZeroThe Exponent Zero

For any number a where a ≠ 0 10 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



Example Example The Exponent Zero The Exponent Zero Simplify: a) 12450 b) (−3)0

c) (4w)0 d) (−1)80 e) −80

Solutionsa) 12450 = 1

b) (−3)0 = 1

c) (4w)0 = 1, for any w 0.

d) (−1)80 = (−1)1 = −1

e) −80 is read “the opposite of 80” and is equivalent to (−1)80: −80 = (−1)80 = (−1)1 = −1



The POWER RuleThe POWER Rule

For any number a and any whole numbers m and n

nmnm aa In other Words:

To RAISE a POWER to a POWER, MULTIPLY the exponents and leave the base unchanged



Quick Test of Power RuleQuick Test of Power Rule

Test 632?32 777

494949497 332

nmnm aa

67777777777777 327



Example Example Power Rule Power Rule

Simplify: a) (x3)4 b) (42)8

Solution a) (x3)4 = x34

= x12

Solution b) (42)8 = 428

= 416

Base is x

Base is 4



Raising a Product to a PowerRaising a Product to a Power

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

nnn baba In other Words:

To RAISE A PRODUCT to a POWER, RAISE Each Factor to that POWER



Quick Test of Product to PowerQuick Test of Product to Power

Test 33?

3 112112

1064822222222112 33

1064813318112 33

nnn baba



Example Example Product to Power Product to Power

Simplify: a) (3x)4 b) (−2x3)2

c) (a2b3)7(a4b5) Solutions

a) (3x)4 = 34x4 = 81x4

b) (−2x3)2 = (−2)2(x3)2 = (−1)2(2)2(x3)2 = 4x6

c) (a2b3)7(a4b5) = (a2)7(b3)7a4b5

= a14b21a4b5 Multiplying exponents

= a18b26 Adding exponents



Raising a Quotient to a PowerRaising a Quotient to a Power

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

n

nn

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 PowerQuick Test of Quotient to Power

Test

3

3?3

7

5

7

5

3

33

7

5

777

555

343

125

7

5

7

5

7

5

7

5

n

nn

b

a

b

a

3

33

7

5

7

5



Example Example Quotient to a Power Quotient to a Power

Simplify: a) b) c) 3

4

w

4

5

3

b

25

4

2a

b

Solution a)3 33

34 644

w w w

4 4

5 45

3 3

( )b b 5 4 20

81 81

b b Solution b)

25 5

4 4

2

2

2 (2 )

( )

a a

b b

2 5 2 10

4 2 8

2 ( ) 4a a

b b Solution c)



Negative Exponents Negative Exponents

Integers as Negative Exponents



Negative ExponentsNegative Exponents

For any real number a that is nonzero and any integer n

nn

aa

1

The numbers a−n and an are thus RECIPROCALS of each other



Example Example Negative Exponents Negative Exponents

Express using POSITIVE exponents, and, if possible, simplify.

a) m–5 b) 5–2 c) (−4)−2 d) xy–1

SOLUTION

a) m–5 =

b) 5–2 =

5

1

m

2

1 1

255



Example Example Negative Exponents Negative Exponents

Express using POSITIVE exponents, and, if possible, simplify.

a) m–5 b) 5–2 c) (−4)−2 d) xy−1

SOLUTION

c) (−4)−2 =

d) xy–1 =

2

1 1 1

( 4)( 4) 16( 4)

1

1 1 xx x

y yy

• Remember PEMDAS



More ExamplesMore Examples

Simplify. Do NOT use NEGATIVE exponents in the answer.a) b) (x4)3 c) (3a2b4)3

d) e) f)

Solution

a)

5 3w w5

6

a

a

9

1

b

7

6

w

z

5 3w w 5 ( 3 2)w w



More ExamplesMore Examples

Solution

b) (x−4)−3 = x(−4)(−3) = x12

c) (3a2b−4)3 = 33(a2)3(b−4)3

= 27 a6b−12 =

d)

6

12

27a

b515 ( )

66a

a a aa

e) ( 9 99

)1b b

b

f) 7 6

7 66 6 7 7

1 1w zw z

z z w w



Factors & Negative ExponentsFactors & Negative Exponents

For any nonzero real numbers a and b and any integers m and n

n

m

m

n

a

b

b

a

A factor can be moved to the other side of the fraction bar if the sign of the exponent is changed



Examples Examples Flippers Flippers

Simplify 6

3 4

20

4

x

y z

SOLUTION We can move the negative factors to

the other side of the fraction bar if we change the sign of each exponent.

3

4

34

6

6

20 5

4

x z

y z y z

6x



Reciprocals & Negative ExponentsReciprocals & Negative Exponents

For any nonzero real numbers a and b and any integer n

nn

a

b

b

a

Any base to a power is equal to the reciprocal of the base raised to the opposite power



Examples Examples Flippers Flippers

Simplify

SOLUTION

24

3

a

b

4

4

223

3

ba

ab

2

4 2

(3 )

( )

b

a

2 2

8

2

8

3 9b b

a a



Summary – Exponent PropertiesSummary – Exponent Properties1 as an exponent a1 = a

0 as an exponent a0 = 1

Negative Exponents(flippers)

The Product Rule

The Quotient Rule

The Power Rule (am)n = amn

The Product to a Power Rule

(ab)n = anbn

The Quotient to a Power Rule

.n n

n

a a

b b

.m

m nn

aa

a

.m n m na a a

1, ,

n nn mn

n m n

a b a ba

b aa b a

This sum

mary assum

es that no denom

inators are 0 and that 00 is not

considered. For any integers m

and n



WhiteBoard WorkWhiteBoard Work

Problems From §1.6 Exercise Set• 14, 24, 52, 70, 84, 92, 112, 130

Base & Exponent →Which is Which?



All Done for TodayAll Done for Today

AstronomicalUnit(AU)



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

AppendiAppendixx

–

srsrsr 22

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

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