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

Oct 20, 2016

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.

$\left(a . b c \ldots \times {10}^{n}\right) \cdot \left(e . f g \ldots \times {10}^{m}\right) = \left(a . b c \ldots \cdot e . f g \ldots\right) \times \left({10}^{n} \cdot {10}^{m}\right)$

In the concrete case we have here:

$2 \times {10}^{- 4} \cdot 3.4 \times {10}^{6} = \left(2 \cdot 3.4\right) \times \left({10}^{-} 4 \cdot {10}^{6}\right)$

The numbers are multiplied:

$\left(2 \cdot 3.4\right) \times \left({10}^{-} 4 \cdot {10}^{6}\right) = 6.8 \times \left(\ldots\right)$

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.: $6.8 \times {10}^{2} = 680$