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so i want to add restrictions to this list of exponent laws. The code i will post now is my attempt at writing their restrictions and the laws i want to write restrictions on. SO, please let me know if my attempt is correct and if it's not i would like to see how to write them correctly.
For all $\mathbf{b,c\in\mathbb{R}}$ and for all $\{a, d,d' \in \mathbb{R} | d,d'> 0\}$ $$d^b\cdot d^c=d^{b+c}$$ $$(d^b)^c=d^{bc}$$ $$(d \cdot d')^c=d^cd'^c$$ $$\left(\frac{d'}{d}\right)^b=\frac{d'^b}{d^b}$$ $$\frac{d^b}{d^c}=d^{b-c}$$ $$d^{-c}=\frac{1}{d^c}$$ $$a^{\frac{b}{d}}=\sqrt[d]{a^b}$$ $$\sqrt[d]{ab}=\sqrt[d]{a}\sqrt[d]{b},\quad\text{where} \textbf{ both } \text{a and b are non-negative}$$ $$\sqrt[d]{\frac{a}{b}}=\frac{\sqrt[d]{a}}{\sqrt[d]{b}},\quad\text{where} \textbf{ both } \text{a and b are non-negative}$$ $$\sqrt[d]{\sqrt[d']{a}}=\sqrt[dd']{a}$$
It is common to define powers $a^b$, $a,b$ real, for $a > 0$ and any $b$. If you consistently restrict all to this case only, I think that all of your rules will be satisfied without any problems.
If you want to allow $a \leq 0$, it's tricky. Sometimes you would like to have things such as $(-8)^{1/3}=-2$, but then if you start working with negative $a$'s, many or all of your other rules will fail; such as $((-4)^2)^{1/2}$ will be defined, but not equal to $(-4)^{2\times \frac{1}{2}}$ and $((-4)^{1/2})^2$ is not defined at all.
The power $a^b$ itself is often not well-defined for negative $a$, not even with complex numbers.