Lesson 2-5

Calculating with Scientific Notation

     Scientific notation is simply a method for expressing, and working with, very large or very small numbers.  It is a short hand method for writing numbers, and an easy method for calculations.  Numbers in scientific notation are made up of three parts: the coefficient, the base and the exponent.  Observe the example below:

5.67 x 10⁵

This is the scientific notation for the standard number, 567 000.  Now look at the number again, with the three parts labeled.
5.67 x 10⁵
        coefficient                          base        exponent

   In order for a number to be in correct scientific notation, the following conditions must be true:

1. The coefficient must be greater than or equal to 1 and less than 10.
2. The base must be 10.
3. The exponent must show the number of decimal places that the decimal needs to be moved to change the number to standard notation.  A negative exponent means that the decimal is moved to the left when changing to standard notation.

Changing numbers from scientific notation to standard notation.

       Ex.1  Change 6.03 x 10⁷ to standard notation.

remember,  10⁷ = 10 x 10 x 10 x 10 x 10 x 10 x 10 = 10 000 000

so,    6.03 x 10⁷ = 6.03 x 10 000 000 = 60 300 000

answer = 60 300 000

Instead of finding the value of the base, we can simply move the decimal seven places to the right because the exponent is 7.

So, 6.03 x 10⁷ = 60 300 000

Now let us try one with a negative exponent.

Ex.2 Change 5.3 x 10^-4 to standard notation.

The exponent tells us to move the decimal four places to the left.

so, 5.3 x 10^-4 = 0.00053

Changing numbers from standard notation to scientific notation

Ex.1  Change 56 760 000 000 to scientific notation

Remember, the decimal is at the end of the final zero.

The decimal must be moved behind the five to ensure that the coefficient is less than 10, but greater than or equal to one.

The coefficient will then read 5.676

The decimal will move 10 places to the left, making the exponent equal to 10.

Answer equals 5.676 x 10¹⁰

Now we try a number that is very small.

Ex.2  Change 0.000000902 to standard notation

The decimal must be moved behind the 9 to ensure a proper coefficient.

The coefficient will be 9.02

The decimal moves seven spaces to the right, making the exponent -7

Answer equals 9.02 x 10^-7 

Calculating with Scientific Notation

     Not only does scientific notation give us a way of writing very large and very small numbers, it allows us to easily do calculations as well.  Calculators are very helpful tools, but unless you can do these calculations without them, you can never check to see if your answers make sense.   Any calculation should be checked using your logic, so don't just assume an answer is correct.  This page will explain the rules for calculating with scientific notation.

Rule for Multiplication - When you multiply numbers with scientific notation, multiply the coefficients together and add the exponents.  The base will remain 10.

Ex 1.  Multiply  (3.45 x 10⁷) x (6.25 x 10⁵)

first rewrite the problem as:    (3.45 x 6.25) x (10⁷ x 10⁵)

Then multiply the coefficients and add the exponents:    21.5625 x 10¹²

Then change to correct scientific notation and round to correct significant digits:  2.16 x 10¹³

NOTE - we add one to the exponent because we moved the decimal one place to the left.

Remember that correct scientific notation has a coefficient that is less than 10, but greater than or equal to one.     

Ex. 2.  Multiply (2.33 x 10^-6) x (8.19 x 10³)

rewrite the problem as: (2.33 x 8.19) x (10^-6 x 10³)

       Then multiply the coefficients and add the exponents:  19.0827 x 10^-3

Then change to correct scientific notation and round to correct significant digits 1.91 x 10^-2

Remember that -3 + 1 = -2

Rule for Division - When dividing with scientific notation, divide the coefficients and subtract the exponents.  The base will remain 10.

Ex. 1 Divide 3.5 x 10⁸ by 6.6 x 10⁴

rewrite the problem as:                 3.5 x 10⁸
                                                 ---------
                                                          6.6 x 10⁴        

Divide the coefficients and subtract the exponents to get:      0.530303 x 10⁴

Change to correct scientific notation and round to correct significant digits to get: 5.3 x 10³

Note - We subtract one from the exponent because we moved the decimal one place to the right.

Rule for Addition and Subtraction - when adding or subtracting in scientific notation, you must express the numbers as the same power of 10.  This will often involve changing the decimal place of the coefficient.

Ex. 1  Add 3.76 x 10⁴ and 5.5 x 10²

move the decimal to change 5.5 x 10² to 0.055 x 10⁴

add the coefficients and leave the base and exponent the same:  3.76 + 0.055 = 3.815 x 10⁴

following the rules for rounding, our final answer is 3.815 x 10⁴        

Rounding is a little bit different because each digit shown in the original problem must be considered significant, regardless of where it ends up in the answer.

Ex. 2  Subtract (4.8 x 10⁵) - (9.7 x 10⁴)

move the decimal to change 9.7 x 10⁴ to 0.97 x 10⁵

subtract the coefficients and leave the base and exponent the same:  4.8 - 0.97 = 3.83 x 10⁵

round to correct number of significant digits: 3.83 x 10⁵