Powers with integer exponents

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1 Lüneburg, Fragment Powers with integer exponents -E

2 -E2

3 What should we know the properties of exponents, the scientific notation of real numbers, power rules. -E3

4 Why should we learn to use powers? Real numbers and algebraic expressions are often written with exponents. In this section we show, how such numbers, as for example M Earth kg, which describes the mass of the Earth, and me kg, which describes the electron mass, can be written in compact form: a 0 m, < a < 0, where m is an integer. -E4

5 Powers as a tool to simplify mathematical expressions Mathematics is sometimes quite complicated, but it is one of the tasks of mathematics to provide tools to simplify long and cumbersome expressions. One of these tools are powers. They are nothing else than a shorthand notation of some multiplications. For example, a repeated multiplication can be written in exponential form: Repeated multiplication: b b b b b4 (5 x ) (5 x) (5 x ) (5 x) 3 ( 3) ( 3) ( 3) ( 3) ( 3) Exponential form: ( 3) 5 ( ) 6 2

6 Integer Exponent Definition: We call the product of n equal factors b, n-th power of b, or b to the power of n bn = b b b... b, n ℕ { 0, }, b ℝ n times 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 and exponent n: p = bn The exponent of a number b says how many times the number is used in a multiplication. Examples: 3 2 = 2 2 2, 6 5 = , 0 3 = =.000, -2a () 7 4 = = =

7 Exponentiation Fig. -: Illustration of an exponentiation -2b

8 Powers of 0, scientific notation Powers of 0 are very efficient in writing large numbers and calculating with them. Instead of writing numbers with a lot of zeros, as for example , we write = = = The form, the number is written down, is called scientific notation or standard form. The scientific notation for a number has the form a 0 m, 2- < a < 0, m ℤ.

Physikal parameters in scientific notation: Example M Earth = 6 0 24 kg Fig. -2: The Earth (http://wallpis.

9 Physikal parameters in scientific notation: Example M Earth = kg Fig. -2: The Earth ( The mass of the Earth is M Earth = = = kg 2-2a 24 times

Physikal parameters in scientific notation: Example 2 M Saturn = 5.69 0 26 kg http://www.verlag.digi-art.de/archiv/album/kosmos/slides/saturn.jpg Fig.

10 Physikal parameters in scientific notation: Example 2 M Saturn = kg Fig. -3: The Saturn The Saturn, the second largest planet of the Solar System, is over 95 times as massive as the Earth. Its mass is M Saturn = 2-2b = kg 95 M Earth

Physikal parameters in scientific notation: Example 3 http://en.wikipedia.org/wiki/saturn Fig.

11 Physikal parameters in scientific notation: Example 3 Fig. -4: The Saturn and the Earth Average distance from Earth to Saturn: 2-2c d.43 billion km = km = km

Physikal parameters in scientific notation: Example 4 d =.5 0 8 km Fig. -5: The solar system (http://wallpaperscrunch.

12 Physikal parameters in scientific notation: Example 4 d = km Fig. -5: The solar system ( The average distance d from the Earth to the Sun is approximately 50 million kilometers. d 50 million km = km = km 2-2d

13 notation notation: or standard Tasks -4form Task : Write each number in scientific notation: a ) , b ) c ) , d ) Task 2: In one year there are hours or minutes. Write these numbers in scientific notation. Task 3: An asian elephant in Hagenbeck zoo in Hamburg has a weight of kg. Write down its weight in scientific notation. Task 4: Blue whales from the Northern Atlantic and Pacific have weights of about 70 tons and lengths of about 27 meters. Write their weight in kilos and the length in centimeters in scientific notation. 2-3a

14 notation notation: or standard Tasks 5-7form Task 5: Write the mass of the Sun in scientific notation: M Sun = kg Task 6: A light-year, i.e 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 3 million years ago. Write down this time in scientific notation. 2-3b

15 notation notation:or Solutions standard, form 2 Solution : a ) = b ) =.43 0 c ) =.0 00 d ) = Solution 2: = h = min 2-4a

16 notation notation: or standard Solution form 3 Fig. -6: Elephant in Hagenbeck zoo, Hamburg The weight of an asian elephant in Hagenbeck zoo: 5400 = kg = 5.4 t t =.000 kg 2-4b

notation notation: or standard Solution form 4 https://en.wikipedia.org/wiki/file:blue_whale_tail.jpg Fig.

17 notation notation: or standard Solution form 4 Fig. -7: Blue whale The weight of a blue whale is about 70 tons. The length is about 27 meters: 70 t = = =,7 0 5 kg 2-4c 27 m = = cm

notation notation:or Solutions standard 5, form 6 https://johnosullivan.files.wordpress.com/202//sun-heats-earth-on-one-hemisphere-only.jpg Fig.

18 notation notation:or Solutions standard 5, form 6 Fig. -8: The Sun and the Earth Solution 5: M Sun = kg =, kg Solution 6: 2-4d m = 9,46 05 m

19 notation notation: or standard Solution form 7 Fig. -9: Spinosaurus 3 million years = = years 94 million years = = years 2-4e

20 notation notation: or standard Task 8 form Example: Write a product as a number. Solution: We can work with this product as follows: = ( ) = = Or we can move the decimal point 4 places to the right : Task 8: Write each number in decimal notation: 2-5a a ) , b ) 3, c ) , d ) 4,4 0 7

21 notation notation: or standard Solution form 8 a ) b ) 3, , , c ) = d ) 4,4 0 7 = b

22 2-5c

23 Definitions So far, the power concept has a definite meaning, if n is a natural number larger than. We now extend the definition of powers to exponents with any natural number including n = 0,, such that b = b, b ℝ and for all n : 0n = 0 n 0, n = Definition: Exponent Zero The zeroth power of a nonzero real number is equal to : b 0 =, 3- b ℝ, b 0

24 Powers Powers with negative base are positive when the exponent is even and negative when the exponent is odd. ( b) 2 n = b 2 n, ( b) 2 n+ = b 2 n + Often used special cases are ( ) 2 n =, for example ( ) 2 n+ = ( ) 4 = ( ) ( ) ( ) ( ) = ( ) 5 = ( ) ( ) ( ) ( ) ( ) = ( a) 3 = ( a) ( a) ( a) = a 3 3-2

25 Powers The expressions ( b) n and bn, b >0 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 being even or odd. In the second case, we first build the power and multiply afterwards by = = 2 4 = = = 6 Here 2 is directly to the left of the exponent, meaning that only 2 is raised to the power 4. The minus sign is not raised to the power. Base and exponent of a power can not be interchanged bn nb 3-3

26 Powers 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 product. But it is useful for many problems, to introduce powers with exponents which are 0 or negative integers. Definition: If b is any real number and n is any positive integer, then b n = bn, bn = b n A negative exponent means a division by n factors b, instead of a multiplication. The only restriction, we have on b n is b 0, as we can not divide by zero. Examples: 0 3 = = 4- = 3 = = = 5 =

Powers with negative integer exponents http://programm.ard.de/sendungsbilder/teaser_huge/008/pocutf8_727986448_original_daccord.

27 Powers with negative integer exponents Niels Bohr and his atomic model The electron is a particle with a negative elementary electric charge and a mass me = kg. 3 decimal places 4-2

Powers with negative integer exponents Illustration of alpha decay, a type of radioactive decay in which an atomic nucleus emits an alpha particle Alpha particles (denoted by the first letter in the

28 Powers with negative integer exponents Illustration of alpha decay, a type of radioactive decay in which an atomic nucleus emits an alpha particle Alpha particles (denoted by the first letter in the Greek alphabet, α) consist of two protons and two neutrons bound together into a particle identical to a helium nucleus. Its mass is m = kg. 27 decimal places 4-3

29 notation notation: or standard Tasks 9, 0 form Example of writing a number smaller than in scientific notation: = Task 9: Write each number in scientific notation: a ) b ) c ) Task 0: Write the mass of an electron (in grams) in scientific notation: m e = g 28 decimal places 4-4a

30 notation notation:orsolutions standard9,form 0 Solution 9: a ) = b ) = c ) = Solution 0: m e = g = g 28 decimal places 4-4b

31 notation notation: or Task standard form Example: Write a product as a number. Solution: We can work with this product as follows: ( = ) = = Or we can move the decimal point 3 places to the left : Task : Write each number in decimal notation: 4-5a a ) , b ) 82, c ) , d ) 8, e ) , f ) 0, 0 9

32 notation notation: or standard Solution form Solution : a ) = b ) 82, = c ) = d ) 8, = e ) = = f ) 0, 0 9 = 0, b

33 notation notation: or Tasks standard 2, 3 form Task 2: Write a number in the form: a 0 n, a 0, n ℤ a ) , b ) c ) , d ) Task 3: Write the following numbers in scientific notation: a ) 37, 4-6a b) 48, c) 84, d ) 5.

34 notation notation: orsolutions standard2, form 3 Solution 2: a ) = b ) = c ) = d ) = Solution 3: a ) 3 7 = 287 = 2, b ) 4 8 = = 6, , c ) 8 4 = 4096 = 4, , 0 3 d ) 5 = 6.05 =, , b

35 Powers with negative integer exponents: Tasks 4-6 Task 4: Determine the numerical value of the powers a ) 0.5 2, b ) , c ) Task 5: Determine c a ) c = , 2 b ) c = Task 6: Determine the expressions using the definition of exponent zero 30, 4-7a a0, a b 0, a0 b0, a 0 a b 0 c 0

36 Powers with negative integer exponents: Solution 4 4-7b a ) b ) c ) (0.2) 3 = 2 2 = 4 4 = ( ) 3 5 = = 2 2 = 4 = 4 = ( 5 ) 3 = 4 4 = = 5 3 = 25

37 Powers with negative integer exponents: Solutions 5, 6 Solution 5: a ) 7 0 =, 2 2 = 4, c = b ) 0 2 = 0, 2 = = 4 3 = =, 4 2 = 6, 2 2 = c = = 0 6 Solution 6: 3 0 =, a 0 =, a 0 + b 0 = + = 2, 4-7c = 4 22 = (a b) 0 = a 0 + (a b) 0 + c 0 = + + = 3

38 4-8a

39 4-8b

Scientific Notation. Scientific Notation. Table of Contents. Purpose of Scientific Notation. Can you match these BIG objects to their weights?

Scientific Notation. Scientific Notation. Table of Contents. Purpose of Scientific Notation. Can you match these BIG objects to their weights? Scientific Notation Table of Contents Click on the topic to go to that section The purpose of scientific notation Scientific Notation How to write numbers in scientific notation How to convert between