# How do you simplify (4.65times10^-2)(5times10^6)?

Oct 29, 2017

$2.325 \times {10}^{5}$

#### Explanation:

This is just one of several tricks. To multiply a decimal by 10 is strait forward. So lets see if we can incorporate that.

Not that 5 is the same as $10 \times \frac{1}{2}$

$\textcolor{b l u e}{\text{Just dealing with the numbers}}$

$4.65 \times 5 \textcolor{w h i t e}{\text{ddd")->color(white)("ddd}} 4.65 \times 10 \times \frac{1}{2}$

$\textcolor{w h i t e}{\text{ddddddddd")->color(white)("ddd}} 46.5 \times \frac{1}{2}$

$\textcolor{w h i t e}{\text{ddddddddd")->color(white)("ddd}} 23.25$

color(white)("ddddddddd")->color(white)("ddd"color(green)(2.325xx10)

There should only be one digit to the left of the decimal point and it is not 'allowed' to be 0.
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$\textcolor{b l u e}{\text{Just dealing with the 10's part of } \left(4.65 \times {10}^{- 2}\right) \left(5 \times \times {10}^{6}\right)}$

Believe it or not all we have to do is add the indeces

${10}^{- 2} \times {10}^{6} \textcolor{w h i t e}{\text{d")=color(white)("d")10^(6-2)color(white)("d")=color(white)("d}} \textcolor{g r e e n}{{10}^{4}}$
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$\textcolor{b l u e}{\text{Putting it all together}}$

$\textcolor{g r e e n}{2.325 \times 10 \times {10}^{4}}$

This may be written as:

$2.325 \times {10}^{1} \times {10}^{4}$

$2.325 \times {10}^{1 + 4}$

$2.325 \times {10}^{5}$