I need further explanation on Exponent rules

[celest hatch]

Hey, I am just looking through my proffessors algebra pdf "recap" on first week of class, and I haven't really done an algebra course since high school, except self-teaching this last month. These are the "rules" that he wrote and I just want further explanation on them. Such as, why does the rule only apply if positive (higher than 0?), also does useful property ring a bell to anyone? I can't find it on the internet. Why does 2/7 = 7/2? Would he simply just mean that n/1 is reverse because it went from a division to a multiplication, and is just using a different example to show that point?

[arctic forge]

adonhs

adonhs

[celest hatch]

The second makes complete sense, thank you so much! but im still super confused on the first. Why does (-4) even have a fraction as a exponent?

I get that normally -1 as an exponent means, as you said reversal of 2/7 -> 7/2 or 3/1 to 1/3

from the notes ive read and how much ive been self teaching algebra

[hallow blaze]

a fraction as a division?

i am not following

you mean why we can write fractions in the exponent?

[celest hatch]

sorry i meant exponent!

A fraction as an exponent

[hallow blaze]

oh

yea

basically fractions represent roots

[celest hatch]

ooohh

[arctic forge]

adonhs

[hallow blaze]

And as you may tell this will get some trouble with negative numbers, as we already mentioned, taking the square root or 4th root or even-th root of a negative number goes wrong...

[celest hatch]

I see!!! so going back to the og question, it's no longer an exponant, but a root?

[hallow blaze]

I think this goes honeslty for all roots, even odd ones

[hallow blaze]

it's both

you can just rewrite it

[celest hatch]

I only know very basic info about roots, so i should learn more about that to paint the picture

[hallow blaze]

and depending on the context work on it

[celest hatch]

okokok

well that's good for me rn, thank you so much :)!

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