Why are exponents defined such that $x^2 = x\times x$, rather than $x^2 = x\times |x|$? Doesn't this alternative definition simplify a lot of things? Currently, if you take $x^2=4$, $x$ can equal $2$ or $-2$. But with this alternative definition, it could only equal 2. Additionally, when taking roots, it eliminates the need for "$i$" — for example $\sqrt{-9} = -3$ because $-3^2 = -3\times |-3| = -9$
Why are exponents currently defined the way they are? Am I onto something, or has this already been tried/isn't useful for some reason?
Best Answer
One fundamental rule we want exponents to follow — in fact, arguably the defining property of exponents — is $$ x^a\cdot x^b=x^{a+b} $$
Your proposed version of exponentiation would violate this rule; for example, it would mean that $$ x^2 \cdot x^2 = |x^4| \neq x^4 $$
Following this rule is considered to be more important than defining exponents so that every number has a unique root.