Understanding Rational Exponents with Negative Bases

[bloodasp]

Can anyone point me to a text or link that summarizes the rules when evaluating/simplifying an expression of the form
(a^n)^(1/m) for a < 0. (a^n)^(1/m) yields different answers for a^(n/m) and (a^(1/m))^n.

Ex:

(-8)^(2/6) = (-8)^(1/3) = -2
(-8)^(2/6) = ((-8)^2)^(1/6) = 2
(-8)^(2/6) = (-8^(1/6))^2 = undefined

Thank you very much!

 

[HallsofIvy]

The general function, a^x, is only defined for positive a.
There simply are no ways of giving "rules" for such manipulation with a negative.

 

[bloodasp]

Thanks HallsofIvy

i've read in some book that
(a^n)^{1/m}
where a < 0, n and m are positive even integers and a^{1/m} is defined can be simplified to
|a|^{n/m}

There are just a few examples on this subject that's why I'm looking for other resources. I've solved several exercises and I different answers. I miss out on when to place the absolute value bars and when not to. As I understand it, the absolute value bars may be removed if n/m always yields a positive value for |a|^{n/m}, otherwise, the absolute value bars must be retained.

I know this is elementary for you guys.

 

[Integral]

Just a LaTex note. surround your exponent with curly brackets {} to get it all elevated.\

a ^{ \frac 1 m}

click on the equation to see the code.

 

[bloodasp]

Just a rewrite

Can anyone point me to a text or link that summarizes the rules when evaluating/simplifying an expression of the form
(a^n)^(1/m) for a < 0. (a^n)^(1/m) yields different answers for a^(n/m) and (a^(1/m))^n.

Ex:

(-8)^(2/6) = (-8)^(1/3) = -2
(-8)^(2/6) = ((-8)^2)^(1/6) = 2
(-8)^(2/6) = (-8^(1/6))^2 = undefined

Thank you very much!

Can anyone point me to a text or link that summarizes the rules when evaluating/simplifying an expression of the form
(a^n)^{1/m} for a < 0. (a^n)^{1/m} yields different answers for a^{n/m} and (a^{1/m})^n.

Ex:

(-8)^{2/6} = (-8)^{1/3} = -2
(-8)^{2/6} = ((-8)^2)^{1/6} = 2
(-8)^{2/6} = ((-8)^{1/6})^2 = undefined

Thank you very much!

 

[VietDao29]

i've read in some book that
(a^n)^{1/m}
where a < 0, n and m are positive even integers and a^{1/m} is defined can be simplified to
|a|^{n/m}

No, that's not true. There's no such way to simplify that.