Consider the following example, discussed here, which highlights potential contradictions when applying exponent rules to expressions with a negative base and a fractional exponent with an even denominator.
$ -1 = (-1)^1 = (-1)^{\frac{2}{2}} = (-1)^{2 \cdot \frac{1}{2}} = \left((-1)^2\right)^{\frac{1}{2}} = (1)^{\frac{1}{2}} = \sqrt{1} = 1 $
This reveals potential contradictions that arise when the “laws of exponents” are applied without considering domain restrictions. A common approach to resolving this seems to be imposing input and domain conditions for applying exponent rules. For example, the following formulation (included by OP) includes such conditions:
RATIONAL EXPONENTS
$a^{\frac{m}{n}} = \left(\sqrt[n]{a}\right)^m = \sqrt[n]{a^m}$, provided that m and n are integers with no common factors, n > 1, and $\sqrt[n]{a}$ is a real number.
Note that the numerator m of the exponent $\frac{m}{n}$ is the exponent of the radical expression, and the denominator n is the index of the radical. When n is even and a < 0, the symbol $a^{\frac{m}{n}}$ is not a real number.
PROPERTIES OF RATIONAL EXPONENTS
If r and s are rational numbers and a and b are real numbers, then
$a^r \cdot a^s = a^{r+s}$
$(a^r)^s = a^{r \cdot s}$
$(ab)^r = a^r \cdot b^r$
$\frac{a^r}{a^s} = a^{r-s}$
$\left(\frac{a}{b}\right)^r = \frac{a^r}{b^r}$
$a^{-r} = \frac{1}{a^r}$
$\left(\frac{a}{b}\right)^{-r} = \left(\frac{b}{a}\right)^r$
provided that all of the expressions used are defined.
(Emphasis mine. Original source: Ratti & McWaters, Precalculus: a right triangle approach, section P.6):
In the earlier example above, the apparent paradox arises when the power rule is applied in the step $(-1)^{2 \cdot \frac{1}{2}} = \left((-1)^2\right)^{\frac{1}{2}}$ even though the condition $\sqrt[n]{a}$ is a real number condition is not satisfied. Since $\sqrt{-1}$ is not a real number, the power rule should not be applied, thereby resolving the "paradox" by invalidating that step.
One issue with this formulation is that (a) it lacks symbolic rigor, and (b) it does not include imaginary / complex numbers.
A more robust formulation, provided by one of the respondents, is:
$\forall r, s \in \mathbb{Q}: \forall a \in \mathbb{R}: \left[ (a^r \in \mathbb{R} \land a^s \in \mathbb{R}) \Rightarrow (a^r)^s = a^{r \cdot s} \right]$
An expanded version that explains why both $a^r$ and $a^s$ need to be real is:
$\forall r, s \in \mathbb{Q}: \forall a \in \mathbb{R}: \left[ (a^r \in \mathbb{R} \land a^s \in \mathbb{R}) \Rightarrow a^{r \cdot s} = (a^r)^s = (a^s)^r \right]$
However, this formulation also doesn’t cover cases involving complex numbers. Moreover, it excludes scenarios like extracting factors of imaginary numbers, as demonstrated in the following example:
Expressing Square Roots of Negative Numbers as Multiples of i
The imaginary number i is defined as the square root of -1.
$\sqrt{-1} = i$
So, using properties of radicals,
$i^2 = \left(\sqrt{-1}\right)^2 = -1$
We can write the square root of any negative number as a multiple of i. Consider the square root of -49.
$\sqrt{-49} = \sqrt{49 \cdot (-1)} = \sqrt{49} \sqrt{-1} = 7i$
(Source: Algebra and Trigonometry 2e by OpenStax. Section 2.4)
This example relies on applying the product rule where not all expressions are real numbers, which would fail if all expressions were restricted to real numbers. In a sense, the first formulation has more coverage as it allows for a general condition (“provided that all expressions are defined”), which covers this case, even while it lacks the precision and symbolic rigor of the second formulation.
Finally, I should reference this post that makes reference to the "true" definition of exponentiation, which I don't understand yet (as I'm currently studying Algebra). It seems relevant for a fully general set of rules, and I wonder if such a framework requires knowledge beyond Precalculus and Calculus.
Here’s how I would frame my question:
- Algebra-Level Rules: Where can I find a robust set of definitions and rules for exponents, formulated in symbolic form (like above), that provide a clear and precise framework for exponentiation with real numbers at the Algebra level?
- Extension to Imaginary Numbers: How can these rules be extended to include imaginary numbers, ensuring that they cover basic operations like those in the example above?
- Is it necessary to wait until PreCalculus / Calculus in order to understand the full set of rules that cover all cases, including all possible exponential operations with real and complex numbers - such as using the 3rd type of definition above? Or, is there a robust and comprehensive set of rules that can be understood on Algebra level, that covers everything?
