The number of significant digits in an answer to a calculation will depend on the number of significant digits in the given data, as discussed in the rules below. Approximate calculations (orderofmagnitude estimates) always result in answers with only one or two significant digits.
I. When are Digits Significant?
8.200 x 10^{3}
has four significant digits 8.20 x 10^{3} has three significant digits 8.2 x10^{3} has two significant digits 

Examples:
9,683  has four significant digits 
15.60007  has seven significant digits 
0.0005  has one significant digit 
15.0000  has six significant digits 
3.1560  has five significant digits 
II. Rule #1: Significant Digits in Multiplication, Division, Trig. functions, etc.
In a calculation involving multiplication, division, trigonometric functions, etc., the number of significant digits in an answer should equal the least number of significant digits in any one of the numbers being multiplied, divided etc.
Thus in evaluating sin(kx), where k = 0.097 m^{1} {two significant digits} and x = 4.73 m {three significant digits}, the answer should have two significant digits.
Example: 1.23
* 4567.89 1.23 has three significant digits; 4567.89 has six significant
digits. The result will have the smaller of these: three significant digits.
Your calculator produces 5618.5047 as a result; round it to three significant
digits and report Note: some people say that you should not report 5620 as your result as the last zero is ambiguous. I say that it is fine. We'll just assume the worst case and say that there are 3 significant digits. 
Note that counted numbers have
an unlimited number of significant digits.
As an example, if a hair dryer uses 1.20 kW of power, then 2
identical hairdryers use 2.40 kW:
III. Rule #2: Significant Digits in Addition and Subtraction
When quantities are being added or subtracted, the number of decimal places (not significant digits) in the answer should be the same as the least number of decimal places in any of the numbers being added or subtracted.
Example:
5.67 J (two decimal places) 1.1 J (one decimal place) 0.9378 J (four decimal places) 7.7 J (one decimal place)Be especially careful with numbers which are given in scientific notation.
Example:
1.2 + (3.45 x 10^{4})
The best way to solve this problem is to write the numbers in a column in ordinary notation:
1.2 + 0.000345Your calculator returns 1.200345, but only one digit right of the decimal is significant.
You may also convert all numbers into scientific notation with the same exponent:
Example:
(1.23 x 10^{5}) + (4.56 x 10^{6}) + (7.89 x 10^{7})1.23 x 10^5 45.6 x 10^5 + 789. x 10^5 The full answer would be 835.83 x 10^{5}, but the last two digits are not significant.
IMPORTANT: Keep One Extra Digit in Intermediate Answers When doing multistep calculations, keep at least one more significant digit in intermediate results than needed in your final answer. For instance, if a final answer requires two significant digits, then carry at least three significant digits in calculations. If you roundoff all your intermediate answers to only two digits, you are discarding the information contained in the third digit, and as a result the second digit in your final answer might be incorrect. (This phenomenon is known as "roundoff error.") 
The Two Greatest Sins Regarding Significant Digits