Question

Best way to think of negative and fractional exponents.

Answer 1

Been basing my math on these two areas off of pure memory instead of conceptualizing them. If anyone can provide an explanation on these, that they have perhaps applied in their own studies, I would greatly appreciate it.

Answer 2

@satellite73

Answer 3

sure hold on

Answer 4

trying to find a way to upload a pdf

Answer 5

if you want a concept, here is the guiding principle

Answer 6

I don't believe we can upload files yet. Though I'd trust a link from you :)

Answer 7

you want \[\huge b^n\times b^m=b^{n+m}\] to always hold

Answer 8

so what does that make, for example, \[2^0\]? since \[2^0\times 2^1=2^{0+1}=2^1\] that forces \[2^0=1\]

Answer 9

then since \[2^{-2}\times 2^2=2^0=1\] that makes \[2^{-2}=\frac{1}{2^2}\]

Answer 10

oh wow

Answer 11

finally \[2^{\frac{1}{3}}\times 2^{\frac{1}{3}}\times 2^{\frac{1}{3}}=2^1=2\] that forces \[2^{\frac{1}{2}}=\sqrt[3]{2}\]

Answer 12

\[2^(1/2) = \sqrt[2]{2}\] ?

Answer 13

\[2^\frac{ 1 }{ 2 } = \sqrt[2]{2}\]

Answer 14

in general, if you have a function with \[f(x)\times f(y)=f(x+y)\] and say \[f(1)=b\] then you can see using generalizations of what i wrote above, it forces \[f(n)=b^n\\
f(0)=1\\
f(-n)=\frac{1}{b^n}\] and
\[f(\frac{m}{n})=\sqrt[m]{b^n}\]

Answer 15

oops last one was a typo \[f(\frac{m}{n})=\sqrt[n]{b^n}\]

Answer 16

of course the real question is, how do you make sense of this exponential function for inputs that are not rational
for that, you need logarithms

Answer 17

Your illustrations are extremely good. It took me a bit of processing but the concept is definitely clear now. Thank you :)

Answer 18

yw