Physics Day 5

Objectives

• SWBAT do exponential math

• Understand factors of 10

Agenda

Do Now

Notes

Worksheet

HW2 due tonight HW3 Thursday

Kiwi thinking about those Physics experiments with

apples

Day 5

Exponential Math

Exponential Expression

• An exponential expression is:

where is called the base and is called the exponent

• An exponent applies only to what it is immediately adjacent to (what it touches)

• Example:

nana

23x 3 not to x,only to appliesExponent 4m negative not to m, only to appliesExponent

32x (2x) toappliesExponent

Meaning of Exponent

• The meaning of an exponent depends on the type of number it is

• An exponent that is a natural number (1, 2, 3,…) tells how many times to multiply the base by itself

• Examples: 23x 4m

32x

xx3mmmm 1

xxx 222 38xexponentinteger any of meaning learn the willsection wenext In the

Rules of Exponents

• Product Rule: When two exponential expressions with the same base are multiplied, the result is an exponential expression with the same base having an exponent equal to the sum of the two exponents

• Examples:

nmnm aaa

24 33 243 63 47 xx 47x 11x

Rules of Exponents

• Power of a Power Rule: When an exponential expression is raised to a power, the result is an exponential expression with the same base having an exponent equal to the product of the two exponents

• Examples:

mnnm aa

243 243 83

47x 47x

28x

Rules of Exponents

• Power of a Product Rule: When a product is raised to a power, the result is the product of each factor raised to the power

• Examples:

nnn baab

23x 223 x 29x

42y 442 y 416y

Rules of Exponents

• Power of a Quotient Rule: When a quotient is raised to a power, the result is the quotient of the numerator to the power and the denominator to the power

• Example:

n

nn

b

a

b

a

23

x

2

23

x 2

9

x

Rules of Exponents

• Don’t Make Up Your Own Rules

• Many people try to make these rules:

• Proof:

nnn baba

222 2323

nnn baba

!!!!TRUE! NOT

!!!!TRUE! NOT

222 2323

Using Combinations of Rules to Simplify Expression with Exponents• Examples:

43225 pm 128425 pm 128165 pm 12880 pm

3325 yx 9635 yx 96125 yx

232332 32 yxyx 6496 98 yxyx 151072 yx

252

332

3

2

yx

yx

104

96

9

8

yx

yx

y

x

9

8 2

Integer Exponents

• Thus far we have discussed the meaning of an exponent when it is a natural (counting) number: 1, 2, 3, …

• An exponent of this type tells us how many times to multiply the base by itself

• Next we will learn the meaning of zero and negative integer exponents

• Examples: 0532

Integer Exponents

• Before giving the definition of zero and negative integer exponents, consider the pattern: 1624

823 422 221

02

12

22

1

2

1

4

1

8134 2733 932 331

03 13

23

1

3

1

9

1

1

2

1

2

2

1

1

3

1

2

3

1

Definition of Integer Exponents

• The patterns on the previous slide suggest the following definitions:

• These definitions work for any base, , that is not zero:

10 an

n

aa

1

a

05 1 32

3

2

1

8

1

Quotient Rule for Exponential Expressions

• When exponential expressions with the same base are divided, the result is an exponential expression with the same base and an exponent equal to the numerator exponent minus the denominator exponent

Examples:

.

nmn

m

aa

a

7

4

5

5

4

12

x

x

374 55

8412 xx

“Slide Rule” for Exponential Expressions

• When both the numerator and denominator of a fraction are factored then any factor may slide from the top to bottom, or vice versa, by changing the sign on the exponentExample: Use rule to slide all factors to other part of the fraction:

• This rule applies to all types of exponents• Often used to make all exponents positive

sr

nm

dc

banm

sr

ba

dc

Simplify the Expression:(Show answer with positive exponents)

141

23

2

8

yy

yy

141

26

2

8

yy

yy

31

8

2

8

y

y

83

128

yy 11

16

y

Scientific Notation

• A number is written in scientific notation when it is in the form:

Examples:

• Note: When in scientific notation, a single non-zero digit precedes the decimal point

integeran is and 101 where,10 na a n

5102.3 9105342.1

201098.6

Converting from Normal Decimal Notation to Scientific Notation

• Given a decimal number:– Move the decimal to the right of the first non-zero digit

to get the “a”– Count the number of places the decimal was moved

• If it was moved to the right “n” places, use “-n” as the exponent on 10

• If it was moved to the left “n” places, use “n” as the exponent on 10

• Examples:

.

5102.3 9105342.1

201098.6

000,320 left places 5 decimal Move

3420000000015.0 right places 9 decimal Move

000,000,000,000,000,000,698 left places 20 decimal Move

na 10

Converting from Scientific Notation to Decimal Notation

• Given a number in scientific notation:– Move the decimal in “a” to the right “n” places,

if “n” is positive– Move the decimal in “a” to the left “n” places,

if “n” is negative

• Examples:

.

5102.3 9105342.1

201098.6

000,320right places 5 decimal Move

3420000000015.0left places 9 decimal Move

000,000,000,000,000,000,698right places 20 decimal Move

na 10

Applications of Scientific Notation

• Scientific notation is often used in situations where the numbers involved are extremely large or extremely small

• In doing calculations involving multiplication and/or division of numbers in scientific notation it is best to use commutative and associative properties to rearrange and regroup the factors so as to group the “a” factors and powers of 10 separately and to use rules of exponents to end up with an answer in scientific notation

• It is also common to round the answer to the least number of decimals seen in any individual number

Example of Calculations Involving Scientific Notation

• Perform the following calculations, round the answer to the appropriate number of places and in scientific notation

9

205

1053.1

1098.6102.3

9

205

10

1010

53.1

98.62.3

9205 10101059869281.14 341059869281.14

3510459869281.1 35105.1

notation? scientificin put this todo toneed wedoWhat

Reviewnmnm aaa

mnnm aa

n

nn

b

a

b

a

n

nn

aaa

11

nmn

m

aa

a