Lüneburg, Fragment Powers with integer exponents · Powers with negative integer exponents The original definition of powers referred to integer positive exponents only, because

Powers with integer exponents

Precalculus

Lüneburg, Fragment

1-E1

1-E2 Precalculus

Precalculus1-E3

What should we know

● the properties of exponents,

● the scientific notation of real numbers,

● power rules.

Precalculus1-E4

Why should we learn to use powers?

Real numbers and algebraic expressions are often written withexponents. In this section we show, how such numbers, as forexample

M Earth ≃ 6.000 .000 .000 .000 .000 .000 .000 .000 kg ,

which describes the mass of the Earth, and

me ≃ 0.0000000000000000000000000000009 kg ,

which describes the electron mass, can be written in compactform:

a⋅10 m , 1 <∣a ∣< 10,

where m is an integer.

Powers as a tool to simplify mathematical expressions

Mathematics is sometimes quite complicated, but it is one of the tasks ofmathematics to provide tools to simplify long and cumbersome expressions.One of these tools are powers. They are nothing else than a shorthand no-tation of some multiplications.

1-1 Precalculus

For example, a repeated multiplication can be written in exponential form:

Repeated multiplication: Exponential form:

b⋅b⋅b⋅b b 4

(5 x)⋅(5 x)⋅(5 x ) (5 x)3

(−3)⋅(−3)⋅(−3)⋅(−3)⋅(−3) (−3)5

16⋅ 1

6 ( 16 ) 2

Integer Exponent

Definition:

We call the product of n equal factors b, n-th power of b, or b to thepower of n

b is called base, n is called exponent.

The operation to raise a base b to the power n is called exponentiation.Exponentiation is the task to calculate the power for a given base b andexponent n:

b n = b ⋅ b ⋅ b . . . bn times

, n ∈ ℕ ∖ { 0, 1 } , b ∈ ℝ

p = b n

Examples:

2 3 = 2 ⋅ 2 ⋅ 2, 5 6 = 5 ⋅ 5 ⋅ 5 ⋅ 5 ⋅ 5 ⋅ 5, (17 )

4= 1

7⋅ 1

7⋅ 1

7⋅ 1

7

1-2a Precalculus

The exponent of a number b says how many times the number is usedin a multiplication.

10 3 = 10 ⋅ 10 ⋅ 10 = 1.000, 10 5 = 10 ⋅ 10 ⋅ 10⋅ 10 ⋅10 = 100.000

ExponentiationExponentiation

Fig. 1-1: Illustration of an exponentiation

1-2b Precalculus

Powers of 10, scientific notation

“Powers of 10” are very efficient in writing large numbers andcalculating with them. Instead of writing numbers with a lot ofzeros, as for example 190.000, we write

190.000 = 1.9 ⋅ 100.000 = 1.9 ⋅ 10 ⋅ 10 ⋅ 10 ⋅ 10 ⋅10 = 1.9 ⋅10 5 .

The form, the number 190.000 is written down, is called scientificnotation or standard form.

The scientific notation for a number has the form

a⋅10 m , 1 <∣a ∣< 10, m ∈ ℤ .

2-1 Precalculus

The mass of the Earth is

M Earth = 6 ⋅ 10 24 kg

Fig. 1-2: The Earth (http://wallpis.com/wp-content/uploads/2013/07/HD-Planet-Earth-Wallpapers.jpg)

2-2a Precalculus

Physikal parameters in scientific notation: Example 1

= 6 ⋅ 10 ⋅10 ⋅10 ⋅ . . . ⋅ 10 ⋅10 ⋅ 1024 times

= 6 ⋅10 24 kg

M Earth ≃ 6.000.000.000.000.000.000.000.000 =

http://www.verlag.digi-art.de/Archiv/album/Kosmos/slides/Saturn.jpg

The Saturn, the second largest planet of the Solar System, is over 95 timesas massive as the Earth. Its mass is

= 5.69 ⋅10 26 kg ≃ 95 M Earth

M Saturn = 5.69 ⋅ 10 26 kg

2-2b Precalculus

Fig. 1-3: The Saturn

M Saturn ≃ 569.000 .000 .000 .000 .000 .000 .000 .000 =


http://en.wikipedia.org/wiki/Saturn

Average distance from Earth to Saturn:

2-2c Precalculus

Fig. 1-4: The Saturn and the Earth

d ≃1.43 billion km = 1.430 .000 .000 km =1.43⋅10 9 km


The average distance d from the Earth to the Sun is approximately 150 millionkilometers.

d = 1.5 ⋅ 10 8 km

2-2d Precalculus

d ≃ 150 million km = 150.000.000 km = 1.5⋅10 8 km

Fig. 1-5: The solar system (http://wallpaperscrunch.com/wallpapers/1/solar-system-wide.jpg)


Scientific notation or standard formScientific notation: Tasks 1-4

2-3a Precalculus

Task 1: Write each number in scientific notation:

a ) 271.900.000, b ) 143.000 .000 .000

c ) 10.100 .000.000, d ) 8.300 .004.000 .000

Task 2: In one year there are 8765.81 hours or 525949 minutes. Write these numbers in scientific notation.

Task 3: An asian elephant in Hagenbeck zoo in Hamburg has a weight of 54000 kg. Write down its weight in scientific notation.

Task 4: Blue whales from the Northern Atlantic and Pacific have weights of about 170 tons and lengths of about 27 meters. Write their weight in kilos and the length in centimeters in scientific notation.

Scientific notation or standard formScientific notation: Tasks 5-7

2-3b Precalculus

Task 5: Write the mass of the Sun in scientific notation:

M Sun = 1.989.000.000.000.000.000.000.000.000.000 kg

Task 6: A light-year, i.e. 9.461.000.000.000.000 meters, is the distance travelled by light in vacuum in one year. Write this number in scientific notation.

Task 7: Spinosaurus is a dinosaur which lived about 94 to 113 million years ago. Write down this time in scientific notation.

Scientific notation or standard form

a ) 271.900.000 = 2.719⋅10 8

b ) 143.000.000 .000 = 1.43⋅1011

c ) 10.100 .000.000 = 1.01⋅1010

Solution 1:

2-4a Precalculus

Solution 2: 8765.81 = 8.76581⋅10 3 ≃ 8.8⋅10 3 h

d ) 8.300.004 .000 .000 =8.300004⋅10 12 ≃8.3⋅10 12

525949 = 5.25949⋅10 5 ≃ 5.3⋅10 5 min

Scientific notation: Solutions 1, 2


Fig. 1-6: Elephant in Hagenbeck zoo, Hamburg

5400 = 5.4⋅10 3 kg = 5.4 t

The weight of an asian elephant in Hagenbeck zoo:

1 t = 1.000 kg

2-4b Precalculus

Scientific notation: Solution 3


Fig. 1-7: Blue whale

170 t = 170⋅10 3 = 170.000 = 1,7⋅10 5 kg

The weight of a blue whale is about 170 tons. The length is about 27 meters:

27 m = 2.700= 2.7⋅10 3 cm

https://en.wikipedia.org/wiki/File:Blue_whale_tail.JPG

2-4c Precalculus



2-4d Precalculus

Solution 5:

M Sun = 1.989.000.000.000.000.000.000.000.000.000 kg = 1,988⋅1030 kg

Solution 6: 9.461 .000 .000 .000.000 m = 9,461⋅1015 m

https://johnosullivan.files.wordpress.com/2012/11/sun-heats-earth-on-one-hemisphere-only.jpg

Fig. 1-8: The Sun and the Earth


Scientific notation or standard formScientific notation: Solution 7

Fig. 1-9: Spinosaurus

113 million years = 113⋅10 6 = 1.13⋅10 8 years

2-4e Precalculus

94 million years = 94⋅10 6 = 9.4⋅10 7 years

Scientific notation or standard formScientific notation: Task 8

Task 8: Write each number in decimal notation:

a ) 2.19⋅10 3 , b ) −3,027⋅105

c ) 1.001⋅10 6 , d ) 4,14⋅10 7

2-5a Precalculus

Example: Write a product as a number.2.762⋅10 4

We can work with this product as follows:

2.762⋅10 4 = 2.762⋅(10⋅10⋅10⋅10)= 2.762⋅10.000 = 27620

Or we can “move the decimal point 4 places to the right”:

2.762⋅10 4 → 27.62⋅10 3 → 276.2⋅10 2 → 2762⋅10 → 27620

Solution:


2-5b Precalculus

a ) 2.19⋅10 3 → 21.9⋅10 2 → 219⋅10 → 2190

b ) −3,027⋅10 5 → −30,27⋅10 4 → −302,7⋅10 3 → −3027⋅10 2 →

→ −30270⋅10 → −302700

c ) 1.001⋅10 6 = 1001000

d ) 4,14⋅10 7 = 41400000


2-5c Precalculus

So far, the power concept has a definite meaning, if n is anatural number larger than 1. We now extend the definitionof powers to exponents with any natural number includingn = 0, 1, such that

and for all n :

b1 = b , b ∈ ℝ

0 n = 0 n ≠ 0, 1 n = 1

DefinitionsDefinitions

Definition: Exponent Zero

The zeroth power of a nonzero real number is equal to 1:

b0 = 1, b ∈ ℝ , b ≠ 0

3-1 Precalculus

Powers with negative base are positive when the exponent is even andnegative when the exponent is odd.

Often used special cases are

(− b) 2 n = b 2 n , (− b) 2 n+1 = − b 2 n+1

(− 1) 2 n = 1, (− 1) 2 n+1 = − 1

(− 1) 4 = (−1)⋅ (−1) ⋅ (−1)⋅ (−1) = 1

(− 1) 5 = (−1) ⋅(−1)⋅ (−1) ⋅(−1)⋅ (−1) = −1

for example

PowersPowers

3-2 Precalculus

(− a) 3 = (−a) ⋅ (−a) ⋅ (−a) = −a 3

do not mean the same. The sequence of the operations is important.In the first case, we raise the negative base – b to the n-th power. The result is positive or negative depending on the exponent beingeven or odd. In the second case, we first build the power and mul-tiply afterwards by - 1.

Base and exponent of a power can not be interchanged

(− b) n and − b n , b > 0

− 2 4 = − 2⋅− 2⋅− 2⋅− 2 = 2 4 = 16

b n ≠ n b

The expressions

PowersPowers

− 2 4 = −1⋅2⋅2⋅2⋅2 = −16

Here 2 is directly to the left of the exponent, meaning that only2 is raised to the power 4. The minus sign is not raised to thepower.

3-3 Precalculus

Powers with negative integer exponentsPowers with negative integer exponents

The original definition of powers referred to integer positive exponents only,because a number b may appear 3 times, but not (-3) times as factor in a pro-duct. But it is useful for many problems, to introduce powers with exponentswhich are 0 or negative integers.

Definition: If b is any real number and n is any positive integer, then

b−n = 1

b n, b n = 1

b−n

4-1

A negative exponent means a division by n factors b, instead of a multiplication.The only restriction, we have on is b ≠ 0, as we can not divide by zero.b−n

Precalculus

10−3 = 1

10 3= 1

10⋅10⋅10= 1

10⋅ 1

10⋅ 1

10

Examples:

0.0005= 510000

= 510⋅10⋅10⋅10

= 5⋅ 110

⋅ 110

⋅ 110

⋅ 110

= 5⋅10−4


http://programm.ard.de/sendungsbilder/teaser_huge/008/POCUTF8_7217986448_Original_Daccord.JPEG

Niels Bohr and his atomic model

The electron is a particle with a negative elementary electric chargeand a mass

me ≃ 0. 000000000000000000000000000000931 decimal places

= 9⋅10−31 kg .

4-2 Precalculus

4-3

Alpha particles (denoted by the first letter in the Greek alphabet, α) consistof two protons and two neutrons bound together into a particle identical to ahelium nucleus. Its mass is


Illustration of alpha decay, a type of radioactive decay in which an atomicnucleus emits an alpha particle

Precalculus

m ≃ 0. 000000000000000000000000006627 decimal places

= 6.6⋅10−27 kg .

Scientific notation or standard formScientific notation: Tasks 9, 10

Task 9: Write each number in scientific notation:

a ) 0.0000026

Example of writing a number smaller than 1 in scientific notation:

0.000357 = 3.57⋅10−4

b ) 0.000000097

c ) 0.00000000034

Task 10: Write the mass of an electron (in grams) in scientific notation:

me = 0. 000000000000000000000000000928 decimal places

g

4-4a Precalculus

Scientific notation or standard formScientific notation: Solutions 9, 10

Solution 9: a ) 0.0000026= 2.6⋅10−6

b ) 0.000000097 = 9.7⋅10−8

c ) 0.00000000034 = 3.4⋅10−10

Solution 10:

me = 0. 000000000000000000000000000928 decimal places

g = 9⋅10−28 g

4-4b Precalculus


4-5a Precalculus

Task 11: Write each number in decimal notation:

a ) 7.42⋅10−3 , b ) 82,06⋅10−5

c ) 3.705⋅10−6 , d ) −8,03⋅10−12

Example: Write a product as a number.3.004⋅10−3

We can work with this product as follows:

3.004⋅10−3 = 3.004⋅( 110

⋅ 110

⋅ 110 )= 3.004⋅0.001 = 0.003004

Or we can “move the decimal point 3 places to the left”:

3.004⋅10−3 → 0.3004⋅10−2 → 0.03004⋅10−1 → 0.003004

Solution:

e ) 0.025⋅10−4 , f ) −0,111⋅10−9

Scientific notation: Task 11


4-5b

a ) 7.42⋅10−3 = 0.00742

c ) 3.705⋅10−6 = 0.000003705

e ) 0.025⋅10−4 = 2.5⋅10−6 = 0.0000025

b ) 82,06⋅10−5 = 0.0008206

d ) −8,03⋅10−12 =−0.00000000000803

f ) −0,111⋅10−9 =−0,000000000111

Solution 11:


Precalculus

Scientific notation or standard formScientific notation: Tasks 12, 13

Task 12: Write a number in the form: a⋅10 n , 1 ⩽∣a ∣⩽ 10, n ∈ ℤ

a ) 0.0000495, b ) 0.0000003007

c ) −0.0003004, d ) −0.0000000000000000000000135

Task 13: Write the following numbers in scientific notation:

a ) 3 7 , b ) 4 8 , c ) 8 4 , d ) 115 .

4-6a Precalculus


4-6b Precalculus

Solution 12: a ) 0.0000495 = 4.95⋅10−5

b ) 0.0000003007 = 3.007⋅10−7

c ) −0.0003004 =−3.004⋅10−4

d ) −0.0000000000000000000000135=−1.35⋅10−23


Solution 13: a ) 3 7 = 2187 = 2,187⋅10 3

b ) 4 8 = 65.536 = 6,5536⋅10 4 ≃ 6,55⋅10 4

c ) 8 4 = 4096 = 4,096⋅10 3≃ 4,1⋅10 3

d ) 11 5 = 161.051 = 1,61051⋅10 5 ≃ 1,61⋅105

Powers with negative integer exponents: Tasks 14-16

Task 14: Determine the numerical value of the powers

a ) 0.5− 2 , b ) 0.25−4 , c ) 0.2−3

Task 15: Determine c

a ) c = 7 0 2 2 12

− 2−1 − 3, b ) c = 0 2 5 0 4 2 2− 2

Task 16: Determine the expressions using the definition of exponent zero

3 0 , a 0 , a⋅b 0 , a 0 b 0 , a 0 a⋅b 0 c 0

4-7a Precalculus

a ) 0.5−2 = 12 −2

= 1

12 2

= 114

= 4

b ) 0.25−4 = 14 −4

= 1

14 4

= 11

256

= 256

c ) (0.2)−3 = ( 15 )

−3= ( 5−1 )−3

= 5 3 = 125

4-7b

Powers with negative integer exponents: Solution 14

Precalculus

b ) 0 2 = 0, 5 0 = 1, 4 2 = 16, 2−2 = 1

2 2= 1

4

c = 0 2 5 0 4 2 2−2 = 0 1 16 14

= 17.25

a ) 7 0 = 1, 2 2 = 4, 2−1 = 12

c = 7 0 2 2 12

− 2−1 − 3 = 1 4 12

− 12

− 3 = 2

Solution 15:

3 0 = 1, a 0 = 1, (a⋅b) 0 = 1Solution 16:

a 0 + b 0 = 1 + 1 = 2, a 0 + (a⋅b) 0 + c 0 = 1 + 1 + 1 = 3

4-7c

Powers with negative integer exponents: Solutions 15, 16

Precalculus

4-8a Precalculus

4-8b Precalculus

Documents

Lüneburg, Fragment Powers with integer exponents · Powers with negative integer exponents The original definition of powers referred to integer positive exponents only, because