SCIENTIFIC NOTATION: expressing a number in terms of a # times a power of 10.
ex. 20 expressed in scientific notation = 2 x 101
200 = 2 x 102
Terminology
2 x 102
exponent
coefficient
Rules of scientific notation:
when correctly expressed: 1 ≤ coefficient < 10
the value of the exponent = the # of places the decimal is moved in order to
arrive at a coefficient greater than or equal to 1 and less than 10.
ex. 200 = 2 x 102; to get from 200 to 2 the decimal point had to be moved 2 places, therefore
the exponent is 2.
If the decimal is moved to the left then the exponent has a + value -- note the example above.
Conversely, if the decimal must be moved to the right in order to attain a correctly expressed
coefficient then the exponent is negative.
ex. .002 = 2 x 10-3
10-3 = 1/103 = 1/1000
.002 = 2/1000 = 2/103 = 2 x 10-3
If a number is already expressed in scientific notation but the coefficient is
expressed incorrectly,
ex. .02 x 104 or 2000 x 10-6
The following rule is useful: the "increase/decrease rule" - if the coefficient is increased then
the exponent must be decreased, conversely, if the coefficient is decreased then the exponent
must be increased.
Using the above examples:
.02 x 104 expressed correctly = 2 x 10?
the exponent is going to change by a factor of 2 since the decimal was moved 2 places.
the coefficient is greater by a factor of 2, therefore the exponent will be decreased by 2.
consider the 2nd example:
2000 x 10-6 = 2 x 10-9? or 2 x 10-3?
2 x 10-3 because 10-3 is greater than 10-6 and the coefficient has been
decreased therefore the exponent must be increased.
Working with Scientific Notation:
Basic Rule: Do the indicated operation to the coefficient and
"back up one mathematical operation on the exponent"
ex. multiplication in a sense is "high powered" addition so when working with
multiplication and scientific notation, multiply the coefficients and add the exponents.
Addition and subtraction:
(2.0 x 104) + (3.0 x 103)
ex.
coefficients are added (subtracted) and exponents are left unchanged since "backing up
one mathematical process" from addition (subtraction) is to do nothing to the exponent.
In terms of an everyday example: when one multiplies
in. x in. one gets in.2. On the other hand when one
adds 2 in. + 2 in. one gets 4 in. not 4 in.2. That is to say the inches - like the exponent in
the problem at hand - does not change.
but one cannot add apples and oranges without making them the common material.
4 apples + 3 oranges = 7 pcs. of fruit.
when working with scientific notation and addition and subtraction the first thing to do is:
1) express in terms of a common exponent:
change
to
(2.0 x 104) + (3.0 x 103)
2.0 x 104
+ .3 x 104
or
20.0 x 103
+ 3.0 x 103
Among other things when doing addition and subtraction work vertically!!
When working with subtraction, when it is necessary to change one of the #s, always change the
subtrahend (the bottom # in a subtraction problem)
2) Add coefficients and the exponent of the sum (difference) is the same as the exponents of
the #s being added (subtracted).
2.0 x 104
.3 x 104
2.3 x 104
+
or
+
20.0 x 103
3.0 x 103
23 x 103
3) If necessary, convert the coefficient into "correct" scientific notation, that is,
coefficient = 1 < # < 10.
ex.
23 x 103
becomes 2.3 x 104
Multipication:
1) multiply coefficients
2) add exponents (is that "backing up" one math. process?)
3) if necessary convert the coefficient to 1 < # < 10
ex.
(2 x 104)(6 x 10-3) = 12 x 101 = 1.2 x 102
1) 2 x 6 = 12
(coefficients)
2) 4 - 3 = 1
(exponents)
1
3) 12 x 10 = 1.2 x 102
coefficient is decreased, therefore exponent is increased.
Division:
1) Divide coefficients
2) What are we going to do with the exponents? subtract exponents (numerator – denom.) Is
division "high powered" subtraction? Consider 20 - 4 = 5, How many times can 5 be
subtracted from 20?
3) If necessary convert the coefficient to 1 < # < 10
ex.
(3 x 103)/(6 x 102) = .5 x 101 = 5 x 100 *
* (any # to the zero power = 1, therefore
.5 x 101 = 5 x 100 = 5 x 1 = 5
Raising the Power (square, cube etc.)
1) square (cube) coefficient (that is do indicated math operation)
2) multiply exponents(raising the power of a # is "high powered" multipication)
3) if necessary convert the coefficient to 1 < # < 10.
ex.
(4 x 104)2 = 16 x 108 = 1.6 x 109
Decreasing the Power (square root, cube root)
**1) adjust the coefficient and exponent so that the exponent when divided by the index is an
integer. Do so in such a way that to in the answer coefficient = 1 ≤ # < 10.
In order to make this happen in square root problems 1 ≤ √Coefficient < 100 (that is, the
coefficient inside the radical must be between 1 and 100). If this is true then the coefficient in the
answer = 1 ≤ # < 10
With cube root 1 ≤ √Coefficient < 1000 (that is, the coefficient inside the radical must be between
1 and 1000). If this is true then the coefficient in the answer = 1 ≤ # < 10
2) take the root of the coefficient
3) divide the exponent by the index.
ex. 1
______
√4 x 106
=
2 x 103
ex. 2
_______
√4 x 10-5
=
_______
√40 x 10-6
** index of a square root, √ , = 2
index of a cube root, √ , = 3
=
6.32 x 10-3