Thanks
qbkr21
$\displaystyle 1 \text{ Angstom} = 1.0 \times 10^{-10} \text{ meters}$ and $\displaystyle 1 \text{ meter} = 1.0 \times 10^9 \text{ nanometers}$
thus
$\displaystyle 1.4 {\color{red}\text{ Angstoms}} = 1.4 (\underbrace{{\color{red}1.0 \times 10^{-10} \text{ meters}}}_{\text{this replaces Angstoms}}) = 1.4 \times 1.0 \times 10^{-10} \times (\underbrace{{\color{red}1.0 \times 10^9 \text{ nanometers}}}_{\text{this replaces meters}})$
we just keep replacing the units with their equivalents
or, to do it your way:
$\displaystyle 1.4 \text{ Angstom} \times \frac {10^{-10} \text{ meters}}{1 \text{ Angstom}} \times \frac {10^9 \text{nanometers}}{1 \text{ meter}}$
to take something from the top and put it in the bottom, or vice versa, we change all the exponents to their negatives. but that is not necessary here. in the denominator we only have 1's. so we just multiply all the numerators together.
to answer your first question: yes, the negative sign stays on top. it is not necessary to move it to the bottom, so don't worry about it. moving it to the bottom will mess you up
this has meters on the right and (prefix)meters on the left. this table is correct. in what we did, the coefficient of meter was 1. that is not the case in this table. you have to solve for 1 meter to do our problem
example: the table says,
$\displaystyle 1 \text{ nanometer} = 1 \times 10^{-9} \text{ meters}$
this is correct, but we want 1 meter = something nanometers. multiply both sides by $\displaystyle 10^9$ to obtain
$\displaystyle 1 \times 10^9 \text{ nanometers} = 1 \times 10^{-9} \times 10^9 \text{ meters} = 1 \times \times 10^{-9 + 9} \text{ meters} = 1 \text{ meter}$