MTH55_Lec-02_sec_1-6_Exponent_Rules.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE...

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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 Rulenmnm 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

aaa

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

Test 246?

4

6

5555

nmn

m

aaa

5555555555

554

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

xx

7

3

88

14

6

(6 )(6 )yy

7 9

3

64r tr t



Example Example Quotient Rule Quotient Rule Solution a)

99 3

3

x xx

6x

Solution b)7

7 33

8 88

48

Solution c)14

14 6 86

(6 ) (6 ) (6 )(6 )y y yy

Solution d)7 9 7 9

3 3

664 4r t r t

tr t r

7 3 9 41 8

4362

r 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 = 1b) (−3)0 = 1c) (4w)0 = 1, for any w 0. d) (−1)80 = (−1)1 = −1e) −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) Solutionsa) (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

ba

ba

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

75

75

3

33

75

777555

343125

75

75

75

75

n

nn

ba

ba

3

33

75

75



Example Example Quotient to a Power Quotient to a Power Simplify: a) b) c)

3

4w

4

5

3b

25

4

2ab

Solution a)3 33

34 644w w w

4 4

5 45

3 3( )b b

5 4 20

81 81b b Solution b)

25 5

4 4

2

2

2 (2 )( )

a ab b

2 5 2 10

4 2 8

2 ( ) 4a ab 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

SOLUTIONa) m–5 =

b) 5–2 =

5

1m

2

1 1255



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

SOLUTIONc) (−4)−2 =

d) xy–1 =

2

1 1 1( 4)( 4) 16( 4)

1

1 1 xx xy 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

aa

9

1b

7

6

wz

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

27ab5

15 ( )6

6a a a aa

e) ( 9 99

)1 b bb

f) 7 6

7 66 6 7 7

1 1w zw zz 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

ab

ba

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

204x

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 54x z

y z y z

6x



Reciprocals & Negative ExponentsReciprocals & Negative Exponents For any nonzero real numbers a

and b and any integer nnn

ab

ba

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

3ab

4

4

223

3baab

2

4 2

(3 )( )ba

2 2

8

2

8

3 9b ba a



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

0 as an exponent a0 = 1Negative 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

na ab b

.m

m nna aa

.m n m na a a

1 , ,n nn m

nn m n

a b a bab aa b a

This summ

ary assumes that no

denominators are 0 and that 0

0 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

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