# Scientific Notation Handout

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```					                               Scientific Notation

Did you know that:
The speed of light is ~300,000,000 m/sec.
Earth’s mass is ~5,967,360,000,000,000,000,000,000 kg.
An electron’s mass is
~0.0000000000000000000000000000091093822 kg.
Earth’s circumference is ~40,000,000 m.

*Standard notation is the normal notation we use on a daily basis.
Ex. 25, 36.23, or 65,000,000

Scientific notation is a way to express very large or very small numbers.
Scientific notation is written as a product of a number (a, which is an integer
or decimal) and a power (b) of 10. The number (a) is always > 1 or < 10.
a10b

NOTE: There is only one digit to the left of the decimal in scientific
notation.

Standard Notation                            Scientific Notation
Speed of light          300,000,000 m/sec                            3.0 108 m/sec
Earth’s Mass            5,967,360,000,000,000,000,000,000 kg         5.96736 1024 kg
Electron’s Mass         0.0000000000000000000000000000091093822 kg   9.1093822 1031
kg
Earth’s Circumference   40,000,00 m                                  4.0 107 m

Rules for writing standard notation to scientific notation.

If you move the decimal to the left, you add the number of times you moved
the decimal to the left to the power.

7,234  7,234 100  7.234 103
*remember that any number to the 0 power is 1, so 100 = 1

If you move the decimal to the right, you subtract the number of times you
moved the decimal to the right from the power.

0.23  0.23100  2.3101
How to write standard notation to scientific notation.

Step 1) Rewrite the standard notation with a power of 10,
Step 2) Move the decimal to its correct position for scientific notation.
Recall the rules for moving the decimal and what to do with the
power of 10.

Ex: 1) 6,703

Step 1) 6,703 100

Step 2) move the decimal three times to the left and add 3 to the power of
10.
6.7031003  6.703103

**Note: if the number (a) gets larger the power (b) must get smaller. If the
number (a) gets smaller the power (b) must get larger.

How to write scientific notation to standard notation.

The rules are still the same for moving the decimal. We want to make the
power of 10 a 0. *Remember that 100  1

Ex: 7.6035 104

The power of 10 is -4. The additive inverse of -4 is +4, so we need to add 4
to the power of 10. Recall, if we add to the power of 10, we need to move
the decimal four times to the left.

7.6035 104  0.00076035 1044  0.00076035

***If the exponent is positive, the number (a) must get larger when writing
scientific notation to standard. If the power is negative the number (a) must
get smaller when writing scientific notation to standard notation.
Multiplying and Dividing Scientific Notation

When multiplying and dividing scientific notation, we will use the
commutative property of multiplication. We will also use our rules for
multiplying and dividing exponents with same bases.

Ex: (7.34102 )(3.51103 )

Step 1) Rewrite using the commutative property of multiplication.

(7.34102 )(3.51103 )  (7.34)(3.51)  (102 )(103 )

Step 2) Multiply the constants and the same bases.

(7.34)(3.51)  (102 )(103 )  25.76341023  25.7634105

Step 3) Check to see if your answer must be in scientific notation.
25.7634 105 is not greater than 1 and less than 10, so we must use
our rules to make the number (a) >1 and <10.

25.7634 105  2.57634 1051  2.57634 106

5.52  106
Ex:
2.3  103

Step 1) Rewrite using the commutative property of multiplication.

5.52 106

2.3 103

Step 2) Divide the constants and divide the same bases.

0.24 1063  0.24 103

Step 3) Remember that our answer must be in scientific notation

0.24 103  2.4 1031  2.4 102

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