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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 $\{a, d,d' \in \mathbb{R} | d,d'> 0\}$ $$d^b\cdot d^c=d^{b+c}$$ $$(d^b)^c=d^{bc}$$ $$(d \cdot d')^c=d^cd'^c$$ $$\left(\frac{d'}{d}\right)^b=\frac{d'^b}{d^b}$$ $$\frac{d^b}{d^c}=d^{b-c}$$ $$d^{-c}=\frac{1}{d^c}$$ $$a^{\frac{b}{d}}=\sqrt[d]{a^b}$$ $$\sqrt[d]{ab}=\sqrt[d]{a}\sqrt[d]{b},\quad\text{where} \textbf{ both } \text{a and b are non-negative}$$ $$\sqrt[d]{\frac{a}{b}}=\frac{\sqrt[d]{a}}{\sqrt[d]{b}},\quad\text{where} \textbf{ both } \text{a and b are non-negative}$$ $$\sqrt[d]{\sqrt[d']{a}}=\sqrt[dd']{a}$$

Al3dium GD
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  • The question, in it's current form, lacks clarity. You should explain what you mean by "adding restrictions" and what you are really asking. – Peter Franek Oct 04 '20 at 20:08
  • adding restrictions? like you know some of these laws aren't defined in some places, known as "restrictions". The laws are already there, and i added my attemp at writing the restrictions, i want to see if the restrictions are correct, or i've missed somewhere – Al3dium GD Oct 04 '20 at 20:11
  • Ah, I see. So you want to cross-check if it's correct, or if some assumptions are missing? What are d, d', integers? – Peter Franek Oct 04 '20 at 20:13
  • whoops, no d,d' in R : d,d' =! 0

    (Which is now edited in the question)

     – Al3dium GD Oct 04 '20 at 20:15
  • And yeah, i wanna see if there are any restrictions that are wrong and need anything else or if it's correct, let me know – Al3dium GD Oct 04 '20 at 20:19
  • Ok. Maybe saying it more clearly in the question would help, not to have the question deleted or downvoted, but I understand now what you mean. – Peter Franek Oct 04 '20 at 20:20
  • So already the first line seems problematic: what is $d^b$ for negative $d$ and real $b$, like $(-1)^\pi$? – Peter Franek Oct 04 '20 at 20:22
  • oh is that another form of undefined – Al3dium GD Oct 04 '20 at 21:11
  • that's true, so should how should i try to fix that? – Al3dium GD Oct 04 '20 at 21:15
  • I think you first need explicitely state with $b^b$ is defined and undefined first. Then list these rules. Example if $d < 0$ then $d^b$ is undefined for irrational $b$ and for $b$ rational with even denominator. Also you jump from base $d$ to $a$ when you start doing roots and start assume $d$ is an natural number. – fleablood Oct 04 '20 at 21:59

1 Answers1

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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.

Peter Franek
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