Calculation with powers of ten

When you multiply, add the powers ten

10³×10² = 10³⁺² = 10⁵

10³×10⁻² = 10³⁻² = 10¹=10

When you divide, subtract the power of ten in the denominator from the power of ten in the numerator.

$$\frac{10^3}{10^2} = 10^{3-2} = 10$$
$$\frac{10^3}{10^{-2}} = 10^{3-(-2)} = 10^5$$


When you multiply two numbers in power-of-ten notation (scientific notation), collect the prenumbers and the powers of ten. Multiply the prenumbers and add the exponents of ten. For example

$$\begin{align*}\left( 2.4 \times 10^3 \right) \left( 1.2 \times 10^2 \right) &= 2.4 \times 1.2 \times 10^{3+2} \\ &= 2.88 \times 10^5 \end{align*}$$


Similarly for division...
$$ \begin{align*} \frac{ 2.4 \times 10^3 }{1.2 \times 10^2 } \\ &= \frac{1.4}{ 1.2} \times 10^{3 - 2} \\ &= 2.0 \times 10^1 \\ &= 20. \end{align*}$$


For adding and subtracting you must express both numbers in the same power of ten before adding or subtracting the prenumbers:

$$\begin{align*} 1.2 \times 10^2 + 2.4 \times 10^3 \\ &= 1.2 \times 10^2 + 24 \times 10^2 \\ &= (1.2+24) \times 10^2 \\ &= 25.2 \times 10^2 \end{align*}$$

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Converting units

You can multiply anything by one without changing it. When you have two units, divide the unit you wish to go to by the equivalent in the unit you are converting from.
Since these two are equal, their ratio is one.
$$ 1 \mu\hbox{m} = 10^{-6} \hbox{m}$$
therefore $$1 = \frac{1 \mu\hbox{m}}{10^{-6} \hbox{m}}$$ also $$1 = \frac{10^{-6} \hbox{m}}{1 \mu\hbox{m}}$$

Now you can multiply the value you wish to convert by the ratio, which is equal to one. Make sure the unit you wish to end up with is in the numerator. The denominator unit cancels with the unit you are converting from and you end up with what you want.
$$\begin{align*} 7.9 \times 10^{-5} \hbox{m} \times (1) \\ &= 7.9 \times 10^{-5} \hbox{m} \times \left( \frac{1 \mu\hbox{m}}{10^{-6} \hbox{m}} \right) \\ &=\frac{7.9 \times 10^{-5}}{10^{-6}} \hbox{m} \frac{\mu\hbox{m}}{\hbox{m}} \\ &= 7.9 \times 10^1 \mu\hbox{m} \\ &= 79 \mu\hbox{m} \end{align*}$$



Let's do another example with inches. Now if we know how many mm in an inch, but the value we wish to convert is in meters, we can convert from meters to mm and then from mm to inches by using the trick two times.
1 inch = 25.4 mm, therefore $$ 1 = \frac{1\; \hbox{inch}}{25.4 \;\hbox{mm}} = \frac{25.4 \;\hbox{mm}}{1\; \hbox{inch}} $$
$$\begin{align*} 7.9 \times 10^{-5} \hbox{m}\\ &= 7.9 \times 10^{-5} \hbox{m} \left( \frac{1\; \hbox{inch}}{25.4 \;\hbox{mm}} \right) \left( \frac{10^3\hbox{mm}}{1\; \hbox{m}} \right) \\ &= \frac{7.9 \times 10^{-5} \times 10^3}{25.4} \hbox{m} \frac{\hbox{mm}}{\hbox{m}} \frac{\hbox{inch}}{\hbox{mm}}\\ &=0.31 \times 10^{-2} \hbox{inch} = 3.1 \times 10^{-3}\hbox{inch} \end{align*}$$

That's called 3.1 mil in English.