Alright, 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 $\mathbf{a,b,c\in\mathbb{R}}$, for all $\mathbf{d,d'\neq 0}$ $$d^b\cdot d^c=d^{b+c}$$ $$(d^b)^c=d^{bc}$$ $$(d \cdot d')^c=d^cd'^c$$ $$\left(\frac{d'}{d}\right)^b=\frac{d'^b}{d^b}$$ $$\frac{d^a}{d^b}=d^{a-b}$$ $$d^{-a}=\frac{1}{d^a}$$ $$a^{\frac{b}{d}}=\begin{cases} \sqrt[d]{a^b} & \text{if d is odd}\\ \sqrt[d]{a^b}, & a\geq 0, \text{if d is even} \end{cases}$$ $$\sqrt[d]{ab}=\sqrt[d]{a}\sqrt[d]{b},\quad\text{where} \textbf{ both } \text{a and b are non-negative}$$ $$\sqrt[d]{\frac{a}{d'}}=\frac{\sqrt[d]{a}}{\sqrt[d]{d'}},\quad\text{where} \textbf{ both } \text{a and d' are non-negative}$$ $$\sqrt[d]{\sqrt[d']{a}}=\sqrt[dd']{a}$$
