Question

How do you find the product of `2times10^-4` and `3.4 times 10^6`?

Answer 1

You must multiply numbers and powers of ten separately.

Explanation:

When we have to do operations with numbers in scientific notation, it is different whether to multiply or divide or need to add or subtract.

Multiplication or division numbers in scientific notation is very simple: we do the operation indicated by the decimal part separately from the powers of ten.

`(a.bc... times 10^n) cdot (e.fg... times 10^m) = (a.bc... cdot e.fg...) times (10^n cdot 10^m)`

In the concrete case we have here:

`2 times 10^{- 4} cdot 3.4 times 10^6 = (2 cdot 3.4) times (10^-4 cdot 10^6)`

The numbers are multiplied:

`(2 cdot 3.4) times (10^-4 cdot 10^6) = 6.8 times (...)`

and the powers of ten are also multiplied, but using the properties of the powers. Remember that when we multiply powers of the same base the result is the same base elevated to the sum of the exponents. Then:

`10^-4 cdot 10^6 = 10^{-4+6}=10^2`

Therefore, the operation that asks us the problem statement is:

`2 times 10^-4 cdot 3.4 times 10^6 = 6.8 times 10^2`

I.e.: `6.8 times 10^2=680`