2.8 × 10 raise to the power of 5
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Answer:
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Answer
The first thing to learn is how to convert numbers back and forth between scientific notation and ordinary decimal notation. The expression "10n", where n is a whole number, simply means "10 raised to the nth power," or in other words, a number gotten by using 10 as a factor n times:
105 = 10 x 10 x 10 x 10 x 10 = 100,000 (5 zeros)
108 = 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 100,000,000 (8 zeros)
Notice that the number of zeros in the ordinary decimal expression is exactly equal to the power to which 10 is raised.
If the number is expressed in words, first write it down as an ordinary decimal number and then convert. Thus, "ten million" becomes 10,000,000. There are seven zeros, so in powers of ten notation ten million is written 107.
A number which is some power of 1/10 can also be expressed easily in scientific notation. By definition,
1/10 = 10-1 ("ten to the minus one power")
More generally, the expression "10-n" (where n is a whole number) means ( 1/10 )n. Thus
10-3 = ( 1 / 10 )3 = 1 / ( 10 x 10 x 10) = 1/1000
10-8 = ( 1 / 10 )8 = 1/100,000,000
Scientific notation was invented to help scientists (and science students!)deal with very large and very small numbers, without getting lost in all the zeros. Now answer the following on a separate sheet of paper and check your answers by clicking on "Answers":
First Problem Set
Express 1-6 in scientific notation, and 7-10 in ordinary notation:
1. 100
2. 10,000,000
3. 1 / 10,000
4. one million
5. 1 / 10,000,000
6. one ten millionth
7. 103
8. 10-5
9. 109
10. 1 X 10-2
Answers
What about numbers that are not exact powers on ten, such as 2000, 0.0003, etc.? Actually, they are only a little more complicated to write down than powers of ten. Take 2000 as an example:
2000 = 2 x 1000 = 2 x 103
As another example, take 0.00003, or "three ten-thousandths":
0.0003 = 3 x 1 / 10,000 = 3 x 10-4
There is a simple procedure for getting a decimal number into the "standard form" for scientific notation:
First, write down the number as the number itself times 100. This can be done because 100 equals one, and any number times one equals that number. The number is now in the standard form:
coefficient x 10 exponent
Second, start moving the decimal point in the coefficient to the right or left. For each place you move the decimal place to the left, add 1 to the exponent. For each place you move it to the right, subtract 1 from the exponent. What you are doing is dividing (or multiplying) the coefficient by 10 each time, while at the same time multiplying (or dividing) the exponent term by 10 each time. Since what you 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:
2000 = 2000 x 100= 200 x 101= 20 x 102= 2 x 103
0.0003 = 0.0003 x 100= 0.003 x 10-1 = 0.03 x 10-2 = 0.3 x 10-3= 3 x 10-4
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 example given. We then say that the number is fully in the standard form. You should always express scientific notation numbers in the standard form. Notice that you don't really have to write down each of the steps above; it is enough to count the number of places to move the decimal point and use that number to add or subtract from the exponent. Some examples:
250,000 = 2.5 x 105 5 places to the left
0.000035 = 3.5 x 10-5 5 places to the right
0.00000001 = 1 x 10-8 = 10-8 8 places to the right
Second Problem Set
Express 1-6 in scientific notation, and 7-10 in ordinry notation:
1. 342,000,000
2. 0.000923
3. eight million
4. 0.0000064
5. 47,682
6. 0.0249
7. 4 x 107
8. 3.22 x 10-3
9. 8.4 x 1010
10. 6.33 x 10-6
Answers
The most difficult kind of calculation that can be done with numbers expressed in scientific notation turns out to be addition or subtraction. Multiplication, division, and raising to powers is actually easier. So, we'll deal with these first.
The rule for multiplying two numbers expressed in scientific notation has three steps:
Multiply the coefficients to get the new coefficient.
Add the exponents (watch the signs!) to get the new exponent.
Get the number into the standard form, if needed.
Examples:
(4 x 103) x (2 x 107) = ( 4 x 2 ) x ( 103 + 7 ) = 8 x 1010
(2 x 10-5) x (2.5 x 108) = ( 2 x 2.5 ) x ( 10-5+ 8 ) = 5 x 103
(3 x 10-7) x (3 x 10-8) = ( 3 x 3 ) x ( 10 -7 + (-8) ) = 9 x 10-15
(4 x 107) x (3 x 105) = ( 4 x 3 ) x ( 10 7 + 5 ) = 12 x 1012= 1.2 x 1013
The steps for division are similar:
Divide the coefficients to get the new coefficient
Subtract the "bottom" exponent from the "top" one (really watch the signs!) to get the new exponent.
Get the number into the standard form, if needed.
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