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Many students think

$\sqrt{a} \sqrt{b}=\sqrt{a\ b}$

$\sqrt{a^2}=a$

$\frac{1}{\sqrt{a}}=\sqrt{\frac{1}{a}}$

but none of the above are true when (a) and (b) are negative.

To avoid such problems, students need to learn the properties below are true for any real* (a, b, c) that meet the conditions specified.

$\ a^b a^c = a^{b+c}$ when $a\neq0.$


When $a^c b^c=(a\ b)^c$
$a^c b^c=(a\ b)^c$ when c is an integer.
$a^c b^c=(a\ b)^c$ when (a) or (b) is positive.


When $\left(a^b\right)^c=a^{b c}$
$\left(a^b\right)^c=a^{b c}$ when c is an integer and $a\neq0.$
$\left(a^b\right)^c=a^{b c}$ when a is positive.


$a^c b^{-c}=\frac{a^c}{b^c}$ for all $a, b, c.$


$a^c \left(\frac{1}{b}\right)^c=\left(\frac{a}{b}\right)^c$ when $a$ is positive.


$a^c b^{-c}=\frac{a^c}{b^c}=\left(\frac{a}{b}\right)^c$ when $c$ is an integer and $b$ is positive


* Both -$\infty$ and $\infty$ are excluded because they are not real numbers.
Two properties specify $a\neq0$ to exclude cases such as $\ 0^5\ 0^{-2} \neq 0^{5-2}$ and $\left(0^{-2}\right)^{-3}\neq0^6.$
Students also need to know a few things about when the above properties can be extended to complex numbers, but I didn't account for that above. What books include most of the above properties?

-----Edit------
Actually, I only expect people to learn how some of the above identities extend to complex numbers when they are in certain fields of study. An electrical engineering student should learn some of that before they study Fourier Transforms, and Z-Transforms. However, the properties listed above are only slightly more complicated than what students typically learn in Algebra II and Pre-Calculus. Sadly, some books actually tell you the fallacies above are true. One example is the following from: Ron Larson et al, Algebra II, Maryland Edition, McDougal Little, 2008.
Ron Larson, et al Consider the next example from Daniel Zwillinger, CRC Standard Mathematical Tables and Formulae, CRC Press 1996.
CRC Press
The second example is a math reference for engineers and scientists. It covers elliptical integrals and partial differential equations, but nothing is mentioned about the properties of $a^b$ that leads to complex numbers. Many books only mention how the properties work for integer and rational exponents. What do the authors think people will do when working with approximate exponents having 16 digits of precision? Do they think people will convert the approximate exponents to rational numbers and work the problem from there? Hopefully by the time students take Pre-Calculus they lean that that the properties work in the case of a positive number to a real power.

For those in math intensive fields, the subject should be revisited by the third year at a university. I don't remember what my professors mentioned about the properties of $a^b$, but I can't find a book that covers what I am looking for. I once took a course on functions of complex variables and the text book said nothing about the properties of $a^b$.

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    $\begingroup$ Who are the target audience for the books you are looking for? High school students? Undergraduates? $\endgroup$ – Joel Reyes Noche Mar 16 at 3:18
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    $\begingroup$ I am not sure that I agree with your assertion that every student needs to know when these properties can be extended to the complex numbers, since I am not entirely sure that I agree that every student needs to know much of anything about complex numbers. I think that the more important thing to do in this case is to very carefully emphasize the hypotheses of the theorem(s) at play here. For example, for any positive $a$ and $b$, we have $\sqrt{ab} = \sqrt{a}\sqrt{b}$. Emphasize the necessity of positivity by watching what goes wrong with $\sqrt{-1}\sqrt{-1}$. $\endgroup$ – Xander Henderson Mar 16 at 22:13
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    $\begingroup$ Pretty much every intermediate algebra, college algebra, and precalculus text I've ever taught out of (going back to 1983) gives the appropriate restrictions for these properties to hold. Also, most books I've taught from explicitly state that $\sqrt{x^2} = |x|.$ I could provide you with titles of dozens of such books on my bookshelves (not all of which I've taught from), with specific page references, but I don't see the point, because you could browse the shelves of any college/university library and find the same. Perhaps of more interest might be which texts DON'T do this. $\endgroup$ – Dave L Renfro Mar 17 at 10:23
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    $\begingroup$ Maybe the only books that correctly discuss $a^b$ (in your sense) are textbooks in complex analysis. $\endgroup$ – Gerald Edgar Mar 18 at 10:15
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    $\begingroup$ In your CRC example, we are to assume all quantities are real. For example, with $\sqrt[x]{ab}=\sqrt[x]{a}\sqrt[x]{b}$ we must assume the quantities $\sqrt[x]{a}$ and $\sqrt[x]{b}$ are real. So the text is not incorrect. $\endgroup$ – user52817 Mar 23 at 15:15

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