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?