I am very confused on what negative exponents are, how they're used in real life, how the formula $a^{-b}$ is the same thing as $1/a^b$, and how this formula works. An example I saw was $10^{-3}$:
$$10^0 = 1$$ $$10^{-1} = 1 / 10 = 0.1$$ $$10^{-2} = 0.1/10 = 0.01$$ $$10^{-3} = 0.01/10 = 0.001$$
I understood that you divide $1$ by $10$ three times, but why is it cubed? Meaning why is it $1/*10^3*$?Please answer all my questions in simple explanation please.
$a^{-1}=\dfrac{1}{a}.$ This is more or less a definition. We use "up to minus one" as a synonym for the inverse number, e.g. $2^{-1}=\dfrac{1}{2}.$ The rest comes from multiplications, e.g. $$ 2^{-3}=2^{(-1)\, \cdot \,3}=\left(2^{-1}\right)^3 = \left(\dfrac{1}{2}\right)^3=\dfrac{1}{2}\cdot \dfrac{1}{2}\cdot \dfrac{1}{2}=\dfrac{1}{8} $$
However, in real life, we use prefixes instead of negative exponents. We say milli for a thousandth, which is a multiplication with $\dfrac{1}{1000}=0.001.$ Micro is a millionth, nano is a billionth, and so on. There are some rare occasions when we need other orders than those multiples of $-3.$ E.g., 1 bar is $10^5=10\cdot 10 \cdot 10 \cdot 10 \cdot 10 $ Pascal. If you express this the other way around, then $1 \,\operatorname{Pa}=10^{-5} \,\operatorname{bar}.$ See, the negative exponent marks a division, here by a hundred thousand.
It is only a shortcut for divisions. The most popular shortcut is probably $$ 1\% = 0.01 =\dfrac{1}{100}=\dfrac{1}{10}\cdot\dfrac{1}{10}=10^{-2} $$ It literally means "per hundred [units]" but is better expressed by "times $1/100."$