Many students think

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



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 (a, b, c) that meet the conditions specified.

$\ a^b a^c = a^{b+c}$ when both sides are defined.

$a^c b^c=(a\ b)^c$ when c is an integer and both sides are defined.
$a^c b^c=(a\ b)^c$ when (0 < a < $\infty$) or (0 < b < $\infty$).

$\left(a^b\right)^c=a^{b\ c}$ when c is an integer and both sides are defined.
$\left(a^b\right)^c=a^{b\ c}$ when (0 < a < $\infty$).
$\left(a^b\right)^c=a^{b\ c}$ when (-1 < b $\leq$ 1).

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

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

$a^c b^{-c}=\frac{a^c}{b^c}=a^c \left(\frac{1}{b}\right)^c=\left(\frac{a}{b}\right)^c$ when (0 < b < $\infty$).

The caveat "when both sides are defined", excludes couter examples such as
$\ 0^5\ 0^{-2} \neq 0^{5-2}$ and $\left(0^{-2}\right)^{-3}\neq0^6.$
What books include most of the above properties?

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. The above properties actually apply when the variables are complex numbers. An electrical engineering student should learn most of that before they study Discrete Fourier Transforms, or Z-Transforms. 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? $\endgroup$
    – JRN
    Mar 16, 2019 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, 2019 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. $\endgroup$ Mar 17, 2019 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. $\endgroup$ Mar 18, 2019 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, 2019 at 15:15


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