Scientific Notation Study Guide
What is scientific notation?: Scientific notation is a shorthand method to represent VERY LARGE
numbers or VERY SMALL numbers.
109
0.0000000576 = 5.76 10-8
Ex: 3,400,000,000 = 3.4
General Structure:
d × 10n
d = Decimal – This must be a
number between 1 and 10.
n = Integer – The number of
times the decimal point is
moved. THIS IS NOT THE
NUMBER OF ZEROS!!!
CONVERTING FROM STANDARD FORM TO SCIENTIFIC NOTATION
STEP 1: Move the decimal point as many times as needed to obtain a decimal number between 1 and
10.
3562
3562.0
3.562
STEP 2: The amount of places the decimal was moved becomes the exponent on the 10
Moved the decimal 3 places  103
STEP 3: Put it all together: 3562
=
3.562
103 in scientific notation.
If the decimal point moves left, the exponent on the 10 is
positive; if it moves right the exponent is negative.
CONVERTING FROM SCIENTIFIC NOTATION TO STANDARD FORM
If the n in
is POSITIVE
Move the decimal point to RIGHT to put it back in to standard form.
Ex.
 3 . 7 3 0 0 0 0 0 0 0
 9 . 2 0 0 0
OR
If the n in
is NEGATIVE
Move the decimal point to LEFT to put it back in to standard form.
Ex.
 0 . 0 0 0 0 0 0 0 0 0 0 0 8 . 6

0 . 0 0 0 0 0 4 . 8 2
OR
MULTIPLYING NUMBERS IN SCIENTIFIC NOTATION
(4.6 × 1013) (2.3 × 1018)
STEP 1: Use the commutative property of multiplication to re-write the problem.
(4.6 × 1013) (2.3 × 1018)
4.6 × 1013 × 2.3 × 1018
4.6 × 2.3 × 1013 × 1018
Remember… The commutative property of multiplication allows you to change the order.
Therefore, you can write your decimals together and your powers of ten together.
STEP 2: Use the associative property of multiplication to re-write the problem.
4.6 × 2.3 × 1013 × 1018
(4.6 × 2.3) × (1013 × 1018)
Remember… The associative property of multiplication allows you to group your numbers.
Therefore, you can group your decimals together and your powers of ten together.
STEP 3: Simplify. Multiply the decimals. Use the product law (exponent laws) to multiply the
powers of ten.
(4.6 × 2.3) × (1013 × 1018)
4.6 × 2.3
1013 × 1018
*Line the right most digits of the numbers
up when you multiply. Multiply as if there
was no decimal point.
Product Law – If the bases are the
same (10) then add the exponents.
4.6
× 2.3
1013 + 18 = 1031
1
1
1 3 8
+ 9 2 0
10.5 8
2
10.58 × 1031
* Put the decimal back in at the end. This
is based on the number of decimal places in
the initial problem.
STEP 4: Put the final answer in scientific notation.
10.58 × 1031
1.058 × 1031 × 101
1. 058 × 1031 + 1
Remember…
d × 10n
Use Product law to combine
powers of ten.
d = number between 1-10
FINAL ANSWER:
1.058 × 1032
1.058 × 1032
DIVIDING NUMBERS IN SCIENTIFIC NOTATION
(8.4 × 1045) ÷ (2.1 × 1015)
STEP 1: Re-Write the problem using a fraction bar to show division.
(8.4 × 1045) ÷ (2.1 × 1015)
STEP 2: Separate the problem in to two fractions.
Remember…This works because when we multiply fractions we multiply straight across.
STEP 3: Simplify. Divide the decimals. Use the quotient law (exponent laws) to divide the powers
of ten.
*Using the traditional long division
algorithm…
We must turn the divisor (2.1) in to a
whole number by moving the decimal point
once to the right. If we do it to the
divisor we must do it to the dividend (8.4).
Bring the decimal point straight up and
divide as you normally would without the
decimal point.
4.
2.1 8.4
-8 4
0
Quotient Law – If the bases are the same
(10) then subtract the exponents
(numerator – denominator).
1045 + 15 = 1030
4 × 1030
STEP 4: Be sure that your final answer is in SCIENTIFIC NOTATION (if necessary)!!
Remember…
d × 10n
Use the Product law to combine
powers of ten.
d = number between 1-10
FINAL ANSWER:
4 × 1030
ADDING/SUBTRACTING NUMBERS IN SCIENTIFIC NOTATION
(4.5 × 1014) + (2.3 × 1016)
STEP 1: Factor out the power of 10 with the largest ORDER OF MAGNITUDE (Exponent) using the
product law (exponent laws).
(4.5 × 1014) + (2.3 × 1014 × 102)
1016
The purpose of doing this is so the order of magnitudes are the same. Therefore, we would
change the larger order of magnitude to match the smaller one. Using the exponent product law
1014 × 102 = 1014 + 2 = 1016.
STEP 2: Use the commutative and associative properties to rearrange the problem.
(4.5 × 1014) + (2.3 × 1014 × 102)
4.5 × 1014 + 2.3 × 102 × 1014
COMMUTATIVE PROPERTY
4.5 × 1014 + (2.3 × 102) × 1014
ASSOCIATIVE PROPERTY
Use the commutative property to change the order of the powers of 10 (if necessary – this
depends on how you initially write it). Then use the associative property to group the decimal and
the power of 10 with the smallest order of magnitude together.
STEP 3: Change the new grouping (decimal × power of 10 with the smallest order of magnitude) to
standard form.
4.5 × 1014 + (2.3 × 102) × 1014
4.5 × 1014 + 230 × 1014
Convert to Standard Form by moving the decimal to the
right two times
OR
2.3 × 100 = 230
STEP 4: Use the distributive property to re-write the problem.
4.5 × 1014 + 230 × 1014
(4.5 + 230) × 1014
Because…
(4.5 + 230) × 1014
STEP 5: Add* what is inside the parentheses together.
(4.5
+ 230) × 1014
234.5 × 1014
* The only difference between an addition problem and a subtraction problem is at this point. If
this is a subtraction problem, you must subtract what is inside the parentheses.
STEP 6: Put your answer back in to scientific notation. Use the product law (exponent laws) to do
this.
234.5 × 1014
2.345 × 1014 × 102
1016
Final Answer:
We must make sure that the decimal is between 1
and 10. Therefore, we need to move the decimal to
the right (positive) two times. We must take this in
to account and multiply the problem by 102. Use the
product law to simplify.
(4.5 × 1014) + (2.3 × 1016) =
2.345 × 1016