Scientific notation is needed any time you need to express a number that is very big or very small. Suppose for example you wanted to figure out how many drops of water were in a river 12 km long, 270 m wide, and 38 m deep (assuming one drop is one millilitre). It's much more compact and meaningful to write the answer as roughly 1,23 × 10 − 14 1,23 × 10 − 14 1,23 times 10^{-14} than it is to write 123120000000000. For one thing, the scientific notation is easier to read, and makes it much easier to tell at a glance what the order of magnitude is (rather than counting zeros).

For another, most of the digits in 123120000000000 are completely meaningless (unless your measurements were very precise). For instance, if the exact river length were really 12.123123 km (we just measured it to the nearest kilometre), then correct number of drops would be 124383242000000, and after the first three digits our result of 123120000000000 is quite inaccurate. So it's better to use a notation (like scientific notation) in which you can suppress the inaccurate digits

Count the number of places to move the decimal point to the right or left and write the number like this:

The number 1,350,000,000 can be written as

The number 0.000000000000017 can be written as

State the problem: 7,5×10−11÷4,5×10+207,5×10−11÷4,5×10+20{7,5 times 10^{-11}} div {4,5 times 10^{+20}}

Group the factors: (7,5÷4,5)×(10−11÷10+20)(7,5÷4,5)×(10−11÷10+20)(7,5 div 4,5) times ( 10^{-11} div 10^{+20})

Subtract the exponents: 1,67×10(−11−(+20))1,67×10(−11−(+20))1,67 times 10^(-11 - (+ 20) )

Answer: 1,67×10−311,67×10−311,67 times 10^{-31}

State Problem: (1,5×10−11)×(4,5×10+20)(1,5×10−11)×(4,5×10+20)(1,5 times 10^{-11}) times (4,5 times 10^{+20})

Group the factors: (1,5×4,5)×(10−11×10+20)(1,5×4,5)×(10−11×10+20)(1,5 times 4,5) times (10^{-11} times 10^{+20})

Add the exponents: 6,75×10(−11+20)6,75×10(−11+20)6,75 times 10^(-11 + 20)

Answer: 6,75×1096,75×1096,75 times 10^9