1 Million Divided By 10000
In this section, you will occasionally be asked to respond some questions. Whenever a problem gear up is given, you should reply the questions on a separate sheet of newspaper and then verify your answers past clicking on "Answers."
The first thing to acquire is how to convert numbers dorsum and along between scientific notation and ordinary decimal notation. The expression "10n", where n is a whole number, just ways "10 raised to the nth power," or in other words, a number gotten by using 10 equally a factor n times:
ten8 = x 10 x x 10 x x x ten 10 10 x 10 x 10 = 100,000,000 (8 zeros)
Discover that the number of zeros in the ordinary decimal expression is exactly equal to the power to which x is raised.
If the number is expressed in words, beginning write it downwardly as an ordinary decimal number so convert. Thus, "ten million" becomes 10,000,000. There are seven zeros, then in powers of ten notation ten million is written 107.
A number which is some power of 1/ten can also be expressed easily in scientific notation. Past definition,
More generally, the expression "x-n" (where n is a whole number) means ( 1/10 )n. Thus
10-viii = ( 1 / 10 )8 = ane/100,000,000
Scientific notation was invented to help scientists (and scientific discipline students!)deal with very large and very small numbers, without getting lost in all the zeros. At present answer the following on a carve up sheet of paper and check your answers by clicking on "Answers":
Express 1-half-dozen in scientific note, and 7-10 in ordinary notation:
| i. 100 | ii. ten,000,000 |
| 3. 1 / 10,000 | four. i million |
| 5. 1 / x,000,000 | half dozen. one ten millionth |
| 7. xthree | 8. ten-5 |
| nine. ten9 | 10. 1 Ten 10-2 |
Answers
What well-nigh numbers that are non exact powers on x, such every bit 2000, 0.0003, etc.? Actually, they are only a footling more than complicated to write down than powers of ten. Take 2000 as an case:
Equally another example, take 0.00003, or "three 10-thousandths":
At that place is a simple process for getting a decimal number into the "standard form" for scientific notation:
First, write down the number every bit the number itself times 100. This can exist done because 100 equals one, and any number times i equals that number. The number is now in the standard class:
Second, start moving the decimal signal in the coefficient to the right or left. For each identify y'all move the decimal place to the left, add 1 to the exponent. For each place you move it to the correct, subtract 1 from the exponent. What you are doing is dividing (or multiplying) the coefficient past 10 each time, while at the aforementioned fourth dimension multiplying (or dividing) the exponent term by ten each time. Since what you lot do to the exponent term undoes what you do to the coefficient, the total number does not change.
Some examples will hopefully make it clear:
0.0003 = 0.0003 x ten0= 0.003 x 10-i = 0.03 ten 10-2 = 0.three x 10-3= 3 ten ten-four
You should move the decimal point until there is exactly one nonzero digit to the left of the decimal point, as in the last case of each case given. We and so say that the number is fully in the standard form. You should always limited scientific annotation numbers in the standard form. Detect that you lot don't really have to write down each of the steps above; it is enough to count the number of places to motion the decimal betoken and use that number to add or decrease from the exponent. Some examples:
0.000035 = iii.five ten 10-v 5 places to the correct
0.00000001 = 1 10 10-8 = 10-viii viii places to the correct
Express 1-6 in scientific notation, and 7-10 in ordinry annotation:
| 1. 342,000,000 | 2. 0.000923 |
| 3. eight million | 4. 0.0000064 |
| five. 47,682 | six. 0.0249 |
| 7. iv x 10vii | eight. iii.22 x 10-3 |
| 9. 8.4 x tenten | ten. 6.33 x 10-half-dozen |
Answers
The most difficult kind of calculation that tin be done with numbers expressed in scientific notation turns out to be improver or subtraction. Multiplication, segmentation, and raising to powers is actually easier. Then, we'll deal with these showtime.
The rule for multiplying two numbers expressed in scientific annotation has three steps:
- Multiply the coefficients to get the new coefficient.
- Add together the exponents (scout the signs!) to get the new exponent.
- Get the number into the standard form, if needed.
Examples:
(two x 10-five) ten (2.5 ten 10eight) = ( two 10 2.5 ) ten ( x-5+ 8 ) = 5 ten 10three
(three x ten-seven) x (3 x 10-8) = ( 3 x 3 ) ten ( 10 -7 + (-8) ) = 9 x 10-xv
(4 x 10seven) x (iii ten 105) = ( 4 ten 3 ) ten ( 10 7 + v ) = 12 x 1012= 1.two x 1013
The steps for division are similar:
- Divide the coefficients to go the new coefficient
- Decrease the "bottom" exponent from the "top" i (really watch the signs!) to get the new exponent.
- Get the number into the standard form, if needed.
Some examples:
(9 x x8) / (3 x 10-v) = ( ix / iii ) 10 ( 10 eight - (-five) ) = 3 ten ( 10 eight + 5 ) = three x 10thirteen
(5 x x3) / (2 x x7) = ( v / 2 ) 10 ( ten iii - 7 ) = 2.5 x 10-4
(ii 10 105) / (4 x 102) = ( ii / 4 ) x ( 10 v - 2 ) = 0.5 x ten3= 5 ten 102
If you are given a number in scientific notation to raise to a power, remember that all this means is that it is used equally a cistron that many times. Only write the number downwards equally many times every bit the power to which it is to exist raised, and utilise the rules for multiplication repeatedly.
Example:
(ii x 105)three = ( ii x 2 x 2 ) 10 ( 10v x x5 10 xv)
(2 x 105)three = 8 x ( 10 five + 5 + five ) = 8 ten 1015
In a situation where you have to raise things to a ability and do multiplication or division, always stop raising to the power first, then practice the other operation.50
Case:
(2 ten 2 x 2) x (109 10 10ix x ten9) / (vi x 6) x (ten-2 ten 10-2)
(eight x 10 nine + ix +nine) / (36 10 ten -2 -2)
(8 ten ten27) / (36 x 10-4)
(8/36) ten (10 27 - (-4))
0.22 10 10 31 = ii.ii x 10 30
Calculate the following:
| 1. (seven 10 x6) / (2 x 104) | 2. (ii x xvii) x (4 10 ten-9) |
| 3. (5 x 108) 10 (5 x teniii) | 4. (vi x 10iii)3 / (3 10 106)four |
| 5. (v x 10iii) / (2 x ten3) | six. (3 x 104)2 |
| seven. (4 x 10-vi)3 | 8. (2 x 105) x (half-dozen x 107) / (four x x8) |
Answers
Add-on and subtraction are a little more involved. In that location are four basic steps:
- Discover the number whose exponent is algebraically the smallest (think, negative numbers are algebraically smaller than positive ones, and the "more negative" the number, the smaller information technology is).
- If the exponents of the numbers are not the same, alter the number with the smaller exponent. Exercise this past moving the decimal point of the coefficient of that number to the left, and adding i to the exponent of that number, until the 2 exponents are equal.
- Add together or subtract the coefficients of the two numbers. The outcome is the coefficient of the issue. The exponent is the exponent of the number you did not alter.
- Put the issue in standard form, if necessary.
Examples:
The algebraically smallest exponent is -7, so we change the 2d term:
2 10 10-7 = 0.2 x x-6 The exponents are at present the same
(3 x 10-6) - (0.2 x ten-6) = ( 3 - 0.two ) x 10-6 = ii.viii ten 10-vi
b) (ix.39 10 x5) + (8 x 10three) = (9.39 10 tenv) + (0.08 ten 105) = 9.47 10 10five
In situations where addition and subtraction are mixed with multiplication and division, practice the multiplication and division offset, and then do the addition and subtraction. And don't forget that raising things to powers always takes priority over multiplication and division!
Examples:
(five ten three) x (tenvi x10-3) + (ii.2 x 10v)
(15 x 10vi-3) + (2.2 x x5)
(xv x 10iii) + (two.ii 10 x5)
(0.fifteen x xv) + (2.2 ten 10v)
2.4 x x5
b) (6.3 x ten3) - (4 x x4)3 / (8 x 105)2
(6.three ten 10three) - (4 x 4 x 4 10 teniv 10 104 10 x4) / (8 x 8 x 10five x 10five)
(six.3 x 10three) - (64 10 x12) / (64 x 1010)
(6.3 x 10three) - ( (64 / 64) x 10 12 - 10 ) )
(six.3 ten 10iii) - (1 x 102)
(6.3 x 103) - (0.1 x 103)
6.2 x 103
Calculate the following:
| 1. (5.vii x 106) + (3 x ten5) | two. (4.2 x 10-viii) - (two.3 x 10-viii) |
| iii. (3.8 x x5) - (2.i x 10half dozen) | 4. (7.43 x 10v) + (1.97 10 107) / (2 x 103) |
| 5. (i.35 x 107) + (8 x 105) | half dozen. (half dozen.52 x 10three) - (1.41 x 105) x (2.31 x 10-3) |
| 7. (8.52 ten 10-nine) + (2.16 x 10-9) | eight. (4.73 x 104) + (3.16 x 1011) / (7.4 x teniii)two |
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Updated 8/26/99 By James E. Heath Copyright Ó 1999 Austin Community Higher
1 Million Divided By 10000,
Source: https://www.austincc.edu/jheath/Math/Scinot/scinot.htm
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