MULTIPYING AND DIVIDING WITH SCIENTIFIC NOTATION
SCIENTIFIC NOTATION
Examples of standard notation:
2500
45.60
0.00034
In the examples above, the number 2500 has an understood decimal to
the far right of the number.
Examples of scientific notation:
2.5 x 103
4.560 x 101
3.4 x 10–4
CONVERTING FROM STANDARD TO SCIENTIFIC NOTATION
1. Move the decimal so that there is only one nonzero number to the
left of the decimal. Ex. 832000 becomes 8.32000
2. Then count the number of places the decimal was moved. This
becomes the exponent for the power of 10. In the above example,
the understood decimal is to the right of the last zero. The decimal
was moved 5 places, so the final answer is 8.32 x 105 .
3. If the original number was greater than one, the exponent will be
positive. If the original number was less than one, the exponent will
be negative.
Examples: 0.000005 = 5 x 10-6
687.8 = 6.878 x 102
In the number 4.5 x 106, 4.5 is the coefficient, 10 is the base & 6 is the
exponent.
To multiply numbers using scientific notation, multiply the coefficients then
add the exponents.
EX 1: (2.3 x 104) x ( 3.1 x 103) = (2.3 x 3.1)(4+3) = 7.13 x 107
EX 2: (1.3 x 106) x ( 6.12 x 10-3) = (1.3 x 6.12)(6 + - 3) = 7.956 x 103
EX 3: (3.3 x 105) x (5.1 x 104) = (3.3 x 5.1)(5+4) = 16.83 x 109 =1.683 x 1010
Since numbers written in scientific notation can have only one number to the
left of the decimal, the answer in example 3 is changed to 1.683 by moving the
decimal one place to the left. For each place the decimal is moved to the left,
the exponent must be increased by 1. If you have to move the decimal to the
right, the exponent would decrease.
To divide numbers using scientific notation, divide the coefficients, then
subtract the exponents.
EX:
6.2 x 105
6.2 = 2 x 10(5-3) = 2 x 102
3.1 x 103 = 3.1
Practice: Convert the following to scientific notation
a.
c.
e.
g.
889000000 = __________
0.0000125 = ___________
16700 = ____________
0.0020 = _____________
b. 68140 = ___________
d. 0.006541 = _________
f. 456.23 = __________
h. 231.90 = __________
CONVERTING FROM SCIENTIFIC TO STANDARD NOTATION
1. A positive exponent means move the decimal that many places to the
right.
Ex. 5.21 x 104 becomes 52100
2. A negative exponent means move the decimal that many places to the
left.
Ex 3.16 x 10-4 becomes 0.000316
Practice: Convert the following to standard notation
i. 6.72 x 103 = _________
j. 3.4 x 10–5 = ________
k. 1.256 x 102 = _______
l. 8.90 x 10-3 = ________
When adding and subtracting exponents, watch for negative exponents. In the
previous example if the exponent in the denominator had been a –3 instead of 3,
the final exponent would have been 5 - - 3 = 5 + 3 = 8.
Practice multiplying and dividing the following:
1. (3.42 x 104) (1.29 x 106) =
2. (6.791 x 105) (2.45 x 10-3) =
3. (4.76 x 10-4) (6.32 x 102 ) =
4. (5.32 x 107) (4.15 x 106) =
5.
(24.4 x 109 )
(6.2 x 105) =
6.
(18.9 x 103)
(2.4 x 10-4) =
7.
(36.4 x 10-4 )
(2.2 x 105) =