Scientific notation and very large and very small numbers

Now let's look at an example of a calculation with standard form. 

Suppose you had to multiply $1000000$1000000 by $1000000000$1000000000. Although the multiplication process is quite straightforward (you just add up the total number of zeros), it is very easy to miscount the number of zeros in each number.

Likewise, multiplying $0.000001$0.000001 by $0.000000001$0.000000001 is straightfoward (you just add up the total number of zeros after the decimal points), but again it would be very easy to make an mistake.

This is where standard form comes in handy. 

For example, the first product above can be rewritten as $10^6\times10^9$106×109 and the computation can be carried out by simply summing the exponents using the multiplication law with little chance of making an error, so our answer would be $10^{15}$1015.

Likewise, the second product above can be rewritten as $10^{-6}\times10^{-9}$10−6×10−9. Again, we can easily compute this by adding the powers using the multiplication law, which would give us an answer of $10^{-15}$10−15.

In standard form, numbers are written in the form $a\times10^b$a×10b, where $a$a is a number between $1$1 and $10$10 and $b$b is any integer (positive or negative) that is expressed as an index of $10$10. If you need a refresher on how to multiply or divide by factors of 10, click here.

Remember

• A negative power indicates how many times smaller the $a$a value will be.
• A positive power indicates how many times larger the $a$a value will be.