Hi all I know that $5^1 = 5$, $5^2 = 25$, $5^3 = 125$.
But why is $5^{1.5} = 11.180339887498949$ ?
How did we get the number $11.180339887498949$ ?
Hi all I know that $5^1 = 5$, $5^2 = 25$, $5^3 = 125$.
But why is $5^{1.5} = 11.180339887498949$ ?
How did we get the number $11.180339887498949$ ?
If $a,b$ are positive integers and $x > 0$, then $x^{a/b}$ is defined as $\sqrt[b]{x^a}$, where the $b^\text{th}$ root of $y>0$ is the unique positive real number $r$ such that $r^b = y$. So $5^{1.5}$ = $\sqrt{5^3}$, i.e. it's the number which squared is $125$.
This leaves open many questions such as "why do $n^\text{th}$ roots exist?" and "what about $x^\alpha$ where $\alpha$ is irrational?"
There are no easy answers to those questions which don't involve a first course in real analysis. If you haven't done university level real analysis, you kind of have to take on faith that exponentiation works and obeys the rules given. I know that's a disappointing answer, but it's the only decent answer I have.
$$ 2\cdot2\cdot2=8, $$ so multiplying by $2$ three times is the same as multiplying by $8$,
and multiplying by $8$ one-third of one time is the same as multiplying by $2$.