To begin, we shall take note of one of the rules of exponents which is;
[tex]x^{-a}=\frac{1}{x^a}[/tex]For example;
[tex]10^{-2}=\frac{1}{10^2}=\frac{1}{100}[/tex]We shall now begin solving the given questions. Note that all answers are required in positive exponents. This means we have to simplify as much as possible without having to write the answer as "raised to power..., negative number."
[tex]\begin{gathered} 1\text{ terameter}=10^{12} \\ 1\text{ micrometer}=10^{-6} \\ 1\text{ nanometer}=10^{-9} \end{gathered}[/tex](a)
[tex]\begin{gathered} \text{Product of 1 terameter and 1 micrometer;} \\ 10^{12}\times10^{-6} \\ =10^{12}\times\frac{1}{10^6} \\ =\frac{10^{12}}{10^6} \\ =\frac{1000000000000}{1000000} \\ =10^6m \end{gathered}[/tex](b)
[tex]\begin{gathered} \text{Quotient of 1 terameter and 1 micrometer} \\ 10^{12}\div10^{-6} \\ =10^{12}\div\frac{1}{10^6} \\ =10^{12}\times\frac{10^6}{1} \end{gathered}[/tex]Please take note of this basic procedure when doing calculations. The division sign is changed to multiplication and at the same time, the dividend is inverted (or turned upside down). So we now have 10 raised to power 12 times 10 raised to power 6 over 1, as shown in the equation editor.
[tex]\begin{gathered} 10^{12}\times10^6 \\ =10^{12+6} \\ =10^{18} \\ (\text{That is; 1000000000000 x 1000000} \\ =1000000000000000000 \\ =10^{18}m \end{gathered}[/tex](c)
[tex]\begin{gathered} \text{Product of 1 terameter and }1\text{ nanometer} \\ 10^{12}\times10^{-9} \\ =10^{12}\times\frac{1}{10^9} \\ =\frac{10^{12}}{10^9} \\ =10^{12-9} \\ =10^3m \end{gathered}[/tex](d)
[tex]\begin{gathered} \text{Quotient of 1 terameter and 1 nanometer} \\ 10^{12}\div10^{-9} \\ =10^{12}\div\frac{1}{10^9} \\ =10^{12}\times10^9 \\ =10^{12+9} \\ =10^{21}m \end{gathered}[/tex](e)
[tex]\begin{gathered} \text{Qotient of 1 nanometer and 1 terameter} \\ 10^{-9}\div10^{12} \\ =\frac{1}{10^9}\div10^{12} \\ =\frac{1}{10^9}\times\frac{1}{10^{12}} \\ =\frac{1}{10^{9+12}} \\ =\frac{1}{10^{21}m} \end{gathered}[/tex](f)
[tex]\begin{gathered} \text{Quotient of 1 nanometer and 1 micrometer} \\ 10^{-9}\div10^{-6} \\ =\frac{1}{10^9}\div\frac{1}{10^6} \\ =\frac{1}{10^9}\times\frac{10^6}{1} \\ =\frac{10^6}{10^9} \\ =10^{6-9} \\ =10^{-3} \\ =\frac{1}{10^3}m \end{gathered}[/tex](g)
[tex]\begin{gathered} \text{Product of 1 nanometer and 1 micrometer} \\ 10^{-9}\times10^{-6} \\ =\frac{1}{10^9}\times\frac{1}{10^6} \\ =\frac{1}{10^9\times10^6} \\ =\frac{1}{10^{9+6}} \\ =\frac{1}{10^{15}}m \end{gathered}[/tex]