I am trying to understand the logic of negative exponents and I stumbled upon the video "Negative Exponents" by YouTuber JLC.
He says negative exponents are "defined". But why is it defined the way it is? Is there some kind of international agreement or some kind of international organization that maintains this definition of negative exponents? What if I make my own definition of negative exponents?
Please enlighten me, many thanks.
An expression such as $b^m$ is a notation whose meaning we get to choose. Already a long time ago it became clear that multiplying a number with itself was a common operation so we decided to give it its own notation. Note the following property of exponentiation for positive exponents: If $m$ and $n$ are positive integers, then $$ b^{m + n} = b^m \cdot b^n $$ since on both sides $b$ is multiplied with itself $m + n$ times.
When defining $b^m$ for negative integers $m$, we would like to keep this property of exponentiation for sake of consistency. For example, it should be true that $$ b^1 = b^{2 - 1} = b^2 \cdot b^{-1}. $$ By multiplying both sides of the equality with $\frac{1}{b^2}$, we are forced to define $b^{-1} = \frac{1}{b}$ in order to have an equality. In fact, using similar reasoning we can write $$ b^1 = b^{m + 1 - m} = b^{m + 1} \cdot b^{-m} $$ so that we are forced to define $b^{-m} = \frac{1}{b^m}$ if we want the additive property of exponents to be always valid.
EDIT: By following this line of thought you can come up on your own with the definition of rational exponents as well. You might know (or will learn in the future) that $b^{1/2}$ is used to denote the positive square root of $b$. For example, $4^{1/2} = 2$. The reason this definition is made is because we would like to have $$ b^1 = b^{1/2 + 1/2} = b^{1/2} \cdot b^{1/2} $$ for all positive numbers $b$. But this is the same as saying $b^{1/2}$ is the square root of $b$.