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.