# How do I distribute an exponent

## Powers of ten page 1

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1 powers of ten page 1

3 Introduction On February 17, 2013 AFP dpa reports: Top performance Germany has the fastest supercomputer in Europe According to the Jülich Research Center, Europe's fastest supercomputer Juqueen manages 5.9 petaflops (arithmetic steps per second). The system built by IBM is around times faster than a modern PC. Jülich brain researchers, among others, use the new system to simulate activity in brain structures. (Source: Nuclear Research Center Jülich) In technology and science we always come across either very large numbers or very small numbers. So we find z. E.g. in the above article the indication of 5.9 petaflops. What is behind this name? Well, peta is an abbreviation for the power of ten. The designation 5.9 petaflops corresponds to the number 5.9 10 flops (flops stands for arithmetic steps per second). If we wanted to write out this number, we would have to write the number, which means five quadrillion and 900 trillion. So we can write down numbers with a lot of zeros in a shorter way by using the powers of ten. Example 1: The number should be written as a power of ten. Our number has nine zeros. This number of zeros then becomes the exponent (the exponent) of the base 10, i.e. 10. This means: Now there are also very small numbers such as B. the number 0, which would be pronounced five billionths. Example 2: The number 0 should be written as a power of ten. This number has a total of nine decimal places. The number of decimal places becomes the exponent (the exponent) of the base 10, but with a negative sign, i.e. 10. This is: 0, page 3

4 Syntax, scientific notation The general syntax of a power of ten is: 10; ; (Note: is the set of real numbers, is the set of natural numbers, i.e. all negative and positive integers including zero) If 10 stands alone, then is 1 and is equivalent to For 0: 10 1 For 0: The The decimal point of is shifted to the right by one place. For 0 the following applies: The decimal place of is shifted to the left by places. Example 3: Conversion of powers of ten into decimal numbers: Decimal numbers with the power of ten zero: Convert, 1 10 3, Because of 10 1, the following applies: 5 0.1 3.4 1 Decimal numbers with the power of ten greater than zero: The decimal place (also an imaginary decimal place ) the prefix of the power of ten is shifted to the right as many places as the exponent indicates. Convert, comma to the right: Decimal numbers with the power of ten less than zero: The decimal place (also an imaginary decimal place) of the prefix of the power of ten is shifted to the left as many places as the exponent indicates. Convert, decimal point to the left: 0.02 0,, 0001 0, Example 4: Conversion of decimal numbers into any power of ten In the reverse, we can also convert decimal numbers into powers of 10. If! 0 is a positive real number, the following applies: For 1 and decrease: We move the decimal point from to the left and write 10, where is the number of decimal places shifted. For 1 and increase: We move the decimal point from the right and write 10, where is the number of decimal places moved. For 1 and decrease: We move the decimal point from to the left and write 10, where is the number of decimal places moved. For 1 and increase: We move the decimal point from the right and write 10, where is the number of decimal places moved. page 4

5 Enlarging the pre-number: The pre-number is increased by moving the decimal point to the right and writing the number of decimal places shifted as a negative exponent of 10. Convert, 0005 0,,,,,,,, 16 271,,,,,, Decreasing the prefix: The prefix is ​​decreased by moving the comma to the left and writing the number of decimal places shifted as a positive exponent of 10. Convert by 5 0.5 10 0,,,, 4 10 0,,, 05 0,,,,, 025 1,,,, 16 2,,,, 805 0,,,, Example 5: Converting decimal numbers to Scientific notation As we have seen in Example 4, a real number can be converted into a power of 10 in a variety of ways. Since this leads to a wide variety of representations, it has been agreed on a uniform representation - the so-called scientific notation. The scientific notation means that the real number shown must always have one and only one non-zero digit in front of the comma and all other digits after the comma. The following therefore applies to this representation: 10; & 1; 10 &;,! 0 Conversion of decimal numbers greater than zero: Convert by from 50,, to 5,,, Convert by from 2007, 16 2, to 2,, page 5

6 Conversion of decimal numbers less than zero: Convert from 0,, 801 0,, to 5,,,, Convert from 0,,, to 2,, Conversion of powers of ten into powers of ten scientific: Convert from 0,,,, to,, 2 10 1, 'Convert from 27,,, to 2,, Addition / subtraction of () *) +, otenzen () *) +, oten can be transformed according to the rules of addition or subtraction. In doing so, however, we must ensure that the exponents of the individual () *) +, otences match in sign and number. The following applies: 10 -. / / If. Is !, then / 10 0 must first be converted in such a way that is, or converted in such a way that is. Examples: Do the following additions / subtractions: 3,,, 77311,,,,, 834,, '412,' 10,005412, '' 8,,, 0831,,, 5 10 0,,, ',, ,,,,,,, 8 10 8,,,,,,, 9 10 0,, 2 10 1,2 10 If scientific notation is required for the addition / subtraction in the result, the result must be in the scientific notation can be converted. Examples: Perform the following additions / subtractions and give the result in scientific notation: 3,,, 77311,,, '412,' 10,005412, '41.2 10 0,,, 530,,,,, 130 ,, 8 10 page 6

7 Multiplication / division of () *) +, otenes () *) +, otencies can be transformed according to the rules of multiplication or division, whereby for () *) +, otencies the rule of the 1st and 2nd power law must be observed. The following applies: 10 - / 10 0 / or / 10 0 / 10-0 Examples: Carry out the following multiplications / divisions: 3,,, 77 11,, '46.8 10 4,, 8 4,,,': 10.5 10 '2 ,, 10'1'2 6.0025 8.08 10: 11,, 7, 10 4,:, 5 10 0.5 10 0.5 0,,' ',,, 91 16 ,, 5055 8.8 10: 18,, 1 10: 12,, 001:, 5 10 Here too, if the result is requested in scientific notation, it may have to be converted into scientific notation after the calculation has been carried out. Examples: Carry out the following multiplications / divisions and give the result in scientific notation: 3,,, '4,, 8 10 4,,,, 5 10 0.5 10 0,, 5 10' 18,,, 50553 ,, 1 10: 12,,, 001:, page 7

8 Unit prefixes In everyday life, in science and technology as well as in physics, we often come across units of measure prefixes, for example: B. the meter, the centimeter, the kilometer, the micrometer etc., or the gram, the kilogram, the milligram etc. In physics we find units of measurement such as B. Joule, Kilojoule, Petajoule etc. Resistances are given in Ohm, Kiloohm etc., Capacities in Farad, Nanofarad, Picofarad etc. Behind these resolutions regarding the units of measurement there is nothing other than certain 10) + powers. The following table provides information about which intent belongs to which 10) + power. Large numbers Factor by which the unit is multiplied Prefix Meaning Sign is Exa Trillion Peta Billiard 9 10 Terra Billion: Giga Billion; Mega Million <10 'Kilo Thousand = Hecto Hundred * Deca Ten> Small Numbers Factor with which the unit is multiplied Prefix Meaning Sign is Deci tenths> 0.110 centi hundredths? 0.0110 milli 0.00110 micro millionths A 0, 'nano billionths 0, pico trillionths, 0, femto billiardths B 10 atto trillionths 10 7 page 8

9 Summary of the rules Note: Powers of ten General syntax: 10; ; The following applies to 0: 10 1 The following applies to 0: The decimal point of is shifted to the right by as many places as the number indicates. If that is not enough, the right of is padded with zeros. For 0 the following applies: The decimal place of is shifted to the left by as many places as the number indicates. If that is not enough, the left of is padded with zeros. Scientific notation: 10; C & 1; 9D;,! 0 addition / subtraction 10 -. / / If is !, then / 10 0 must first be converted so that is, or converted so that is. Multiplication / division 10 - / 10 0 /; 10 - / 10 0 / 10-0 page 9

10 Level 1 Basics Sheet 1 Document with 63 questions Exercise A1 Enter the following powers of ten as written out numbers. a) 10 b) 10 c) 10 d) 10 e) 10 f) 10 g) 10 h) 10 Exercise A2 Enter the following numbers in powers of ten. a) 100 b) c) 1000 d) e) 10 f) g) h) 1 Exercise A3 Enter the following powers of ten as written out numbers. a) 10 b) 10 c) 10 d) 10 e) 10 f) 10 g) 10 Exercise A4 Give the following numbers in powers of ten. a) 0.01 b) 0.0001 c) 0.001 d) 0, e) 0.1 f) 0.00001 g) 0, Exercise A5 Give the following arithmetic expressions in whole numbers. a) 1 10 b) 2 10 c) 3 10 d) 4 10 e) 5 10 f) 6 10 g) 7 10 h) 8 10 Exercise A6 Enter the following whole numbers in scientific notation. a) 100 b) c) 3000 d) e) 50 f) g) h) 8 Exercise A7 Give the following whole numbers in scientific notation. a) b) c) d) 700 million e) 0, f) 0, g) 28 micrometers h) 0, Exercise A8 Write as a decimal number. a) 3 10 b) 9.76 10 c) 32.55 10 d) 5.86 10 e) 87.13 10 f) g) 10 h) 0, Exercise A9 Which numbers are the same? Page 10

11 Level 1 Basics Sheet 1 Solution A1 a) 100 b) c) 1000 d) e) 10 f) g) h) 1 Solution A2 a) 10 b) 10 c) 10 d) 10 e) 10 f) 10 g) 10 h) 10 solution A3 a) 0.01 b) 0.0001 c) 0.001 d) 0, e) 0.1 f) 0.00001 g) 0, solution A4 a) 10 b) 10 c) 10 d) 10 e) 10 f) 10 g) 10 solution A5 a) 100 b) c) 3000 d) e) 50 f) g) h) 8 solution A6 a) 1 10 b) 2 10 c) 3 10 d) 4 10 e) 5 10 f) 6 10 g) 7 10 h) 8 10 solution A7 a) 2 10 b) 3.54 10 c) 10 d) 7 10 e) 5 10 f) 1.8 10 g) 2, mh ) 2,3 10 solution A8 a) b) 9760 c) d) e) f) 0.0765 g) 0.00001 h) 0, solution A9 We belong together:,, 1 10 0,, 011 0.0011 0 ,,, 11 10 I don't have a partner Page 11

12 Level 1 Basics Sheet 2 Document with 55 exercises Exercise A1 Calculate without a calculator. a) 1.2 10 4.9 10 b) 3.4 10 2.1 10 c) 2.3 10 3.4 10 d) 3.1 10 1.5 10 e) 7,, 1 10 f), 3 10 g) 3,, 12 10 Exercise A2 Calculate without a calculator. a) b) c) 10:10 d) e) f) g) 2, h): 7 10 i) j) 6 10: 3 10 k) 4.8 10: 1.2 10 Problem A3 What number stands for ? a) b) c) 10: 10 10 d) e) f) 10: 10 1 Exercise A4 What number does it represent? a) b) c) 10 d) e) 10 1 f) 10: 1 Exercise A5 Simplify. a)! "# \$ \$" b) \$ \$ "\$! c)"! # \$% d) # \$! e) 9,, 41 10 f) g) h) 8 10: 2 10 i) j) k) l) 8 10 &: 4 10 Exercise A6 Convert to scientific notation. a) 275 b) c) d) 0.25 e) 0.0004 f) g) 0.2 10 h) i) j) 0, k) 54 micrometers l) 550 million Exercise A7 Which terms are equivalent? page 12

13 Level 1 Basics Sheet 2 Solution A1 a) 6.1 10 b) 1.3 10 c) 5.7 10 d) 3.25 10 or 32.5 10 e) 13.77 10 or 1, f) or 2 , 38 10 g) 3, or 30.22 10 solution A2 a) 10 b) c) 10 d) e) f) g) 7, h) 2 10 i) j) k) solution A3 a) b) c) 10: 10 10 d) e) f) 10: 10 1 solution A4 a) b) c) 10 d) e) f) 10: 10 1 solution A5 a)! "3 10 b) !!!! 10 c) 10 d) !!! 30 10! E) 9,,, 82 10 f) g) 6 10 # h) \$ 8 10%: \$ 2 10% 4 10 i) j) \$ 5 10% \$ 4 10% 20 10 k) l ) \$ 8 10 &%: \$ 4 10% 2 10 Solution A6 a) 2.75 10 b) 9.2 10 c) 2.5 10 'd) 2.5 10 e) 4 10 f) 3.14 10 g) 2 10 h) 5 10 i) 2.53 10 j) 2.7 10 k) 5.4 10 meters l) 5.5 10 Solution A7 We belong together: 10 I have no partner Page 13

14 Level 2 Advanced Sheet 1 Document with 60 questions Exercise A1 Enter the following small numbers as powers of ten in scientific notation. a) 0.0012 b) 0, c) 0.0125 d) e) 0, f) g) h) 0, Exercise A2 Enter the following lengths in meters. Use the scientific notation for this. a) 0, b) 0.012 c) 0, d) 0, e) 0, f) 0.412 g) 0.00086 h) 0.06785 Exercise A3 Write without a power of ten and then convert to the given unit. a) The mass of a queen bee is around 2.5 10. b) The thickness of a bulletproof glass pane is around c) The volume of a dice is around. Thus the queen bee has a mass of approx. The thickness of a bulletproof glass pane is therefore approx. The dice has a volume of approx. Exercise A4 Convert the powers of ten into decimal numbers. a) 5 10 b) 0.1 10 c) 3.4 10 d) 10 e) 3 10 f) g) 0.13 10 h) 10 i) 2 10 j) k) 0.1 10 l) 10 exercise A5 Convert the numbers to scientific notation. a) 50.25 b) 800 c) 2225.88 d) e) 2007.16 f) 2.2805 g) h) 0.05025 i) 0.801 j) 0, k) 17, l) 0, m) 0 , n) 0.5 10 o) 10, p) 0, q) 17, r) 27.16 10 Exercise A6 Carry out the following additions / subtractions. a) 3,, 88 10 b) 6.8 10 4.8 10 c) d) 0,, e) 8,, 92 10 f) 0.5 10 0.5 10 g) h) 18,, 5 10 i) 8.8 10 8.8 10 j) 1.1 10 2.2 10 k) 0, page 14

15 Solution A1 Level 2 Advanced Sheet 1 a) 1.2 10 b) 4, c) 1.25 10 d) 4 10 e) 2 10 f) 1, g) 7.2 10 h) 5 10 Solution A2 a) 2.2 10 b) 1.2 10 c) 6 10 d) 9.98 10 e) 5 10 f) 4.12 10 g) 8.6 10 h) 6, solution A3 a) The queen bee has a mass of 0, that's 0.25. b) The thickness of the armored glass pane is 0.04. That is 4. c) The volume of the dice is 0. That is 1. Solution A4 a) 5 b) 0.1 c) 3.4 d) 1 e) 300 f) g) h) i) 0.02 j ) 0.00036 k) 0.0001 l) 0, solution A5 a) 5, b) 8 10 c) 2, d) 1, e) 2, f) 2.2805 g) 2 10 h) 5, i) 8.01 10 j) 2, k) 1, l) 2, m) 3 10 n) 5 10 o) 1, p) 2.2 10 q) 1, r) 2, solution A6 a) 15.65 10 b) 2 10 c) 5 10 d) e) 10 f) 0.55 10 or 5.5 10 g) 9 10 h) 1890.8351, i) 8, or 888.8 10 j) 8.8 10 k) 2.1 10 page 15

16 Level 2 Advanced Sheet 2 Document with 29 tasks Task A1 Calculate the multiplications / divisions and, if necessary, give the result in scientific notation. a) 3,, 88 10 b) 6.8 10 4.8 10 c) d) 12,: 0.5 10 e) 8.08 10: 1.92 10 f) 10: 10 g) 0.5 10 0.5 10 h) i) 18,, 05 10 j) 8.8 10: 8.8 10 k) 1.1 10: 2.2 10 l) 0.001: 2 10 Exercise A2 Enter in scientific notation. a) Speed ​​of light b) Area of ​​Europe c) Distance from Earth to Sun 150 million d) Age of the universe 13 billion years e) Width of DNA double strand 2.5 nanometers in f) Thickness of aluminum foil 15 in g) 6307 km in h) 0.04 in i) 800 in j) 1 in Exercise A3 How many digits do these numbers have in the decimal system? a) 8.5 billion (estimated number of people living in the year 2025) b) 500 million light years (most distant observed nebulae in the universe) c) How many digits does the number 10 have! "#, if you write it in decimal? Exercise A4 Complete the following table: 0,! \$ 1, Exercise A5 The mass of the earth's moon is about 7, a) Give the mass as a power of ten in tons (%). b) The mass of the earth is about 5, & and is therefore greater than the mass of the moon. How many times the earth is heavier? c) The surface of the moon is about Write this number in and scientific notation. Page 16

17 Solution A1 Level 2 Advanced Sheet 2 a) 44,, b) 32,, c) 6 10 d) 24.012, e) 4, f) 1 g) 0,, 5 10 h) 10 i) 303.50553, j ) k) 5 l) 0, solution A2 a) b), 97 10 c) 150 million 1.5 10 d) 13 billion years 1.3 10 years e) 2.5 nanometers 2.5 10 f) 15 1, 5 10 g), h) 0, i)! j) 1 10 solution A3 a) 8.5 billion digits b) 500 million digits c) 10 \$% & '(10 %% 11 digits solution A%) 1,, 01 0,) Solution A5 a) 7,, * .,! % +, b) !,%, 3. The earth is about 81.3 times heavier than the moon. c), 8 10 page 17

18 Level 3 Expert Sheet 1 Document with 14 tasks Task A1 A very finely knotted oriental silk carpet has a knot density of around 500 knots per square centimeter. a) How many knots does such a carpet of dimensions 2 3 contain? Also give the result in power notation. b) How many knots does the worker have to tie per minute on such a carpet so that a 1 large carpet is finished in one year and the year is assumed to be 1600 hours of work. Exercise A2 Write in the units given in brackets. a) Length of Earth's orbit 9.4 10 () b) Diameter of a cell 20 () c) Distance from Earth to Moon: 3.84 10 () d) Wavelength of blue light: 480 () e) Power of a power plant: 1.8 (f) Atomic diameter: 0.1 () Exercise A3 Important parameters of our solar system are the diameters of the sun, moon and earth: 1.4 10 (sun), 3.48 10 (moon), 1.28 10 (earth ). For an exhibition, the earth is to be represented by a ball with a diameter of 20. Calculate the diameters for the sun and moon in this system.Exercise A4 Atoms have a diameter of about 10! "#. Inside are the atomic nuclei with a diameter of about 10 \$% 1 \$ & 10!". The atomic nucleus has about 99.9% of the mass of the entire atom. a) By what factor is the diameter of the nucleus smaller than that of the entire atom? b) To illustrate the proportions, we imagine the atom as a balloon with a diameter of 10. A small ball inside the balloon is supposed to represent the atomic nucleus. What diameter should it have? c) How much would the little ball have to weigh if balloon 1 weighs? Page 18

19 Level 3 Expert Sheet 1 Task A5 A (water contains about 3.35 10 molecules. The following task shows how unimaginably large this number is At this moment the water molecules would be transformed into grains of sand with a diameter of about 1 and would be distributed evenly over Germany (area approx. 3.5 10). Determine roughly how high Germany would be covered with sand. b) Think of the molecules of 1) colored water and pour this colored water into the North Sea. After a few years, when the colored water has spread well over the oceans, samples are taken from each 1). Can you find at least one colored molecule on average in each sample? (Volume of the world's oceans approx. 1.34 10 () page 19

20 Level 3 Expert Sheet 1 Solution A1 a) Area of ​​the carpet Number of knots in the carpet b) Number of knots per number of knots in the area Working minutes / year, 6 10 Carpet knots / minute 10 52.08, A worker must knot about 52 knots / minute, so that the carpet will be ready in a year. Solution A2 a) 9.4 10 9, b) 20! "0.00002 c) 3,, d)" 0, e) 1.8 # \$% &% '' 1.8 10 ((f) 0.1 0.1 10 "1 10" 0, Solution A3 calculation of the scale 20 0.2 0.2 1,, 256 10 "2.56 10" Calculation of the lunar model) * +, - 3,, 56 10 "8," 8 Calculation of the solar model). + ,, / 1.4 10 2.56 10 "3, Solution A4 a) Calculation of the reduction factor The diameter of the atomic nucleus is about times smaller than that of the entire atom. B) Calculation of the reduction factor The enlargement factor is about 10. Calculation of the diameter of the ball in the balloon" "1 The ball in the balloon has a diameter of about 1. c) Calculation of the ball weight # 456 / \$ 0.999 9.81 /: 9.99: page 20

21 Level 3 Expert Sheet 1 Solution A5 a) Number of water molecules in 1; Water 1; 1000 <,, 35 10 Area of ​​the molecules converted into grains of sand It is assumed that a grain of sand covers an area of ​​1 .. =, - 3,, 35 10 <3,, 35 10