My classmates doesn't understand Fractional and Negative exponents, since I was the top of my class, so they all came to me… Is there any way to explain it clearly to them?
[Math] How to explain Fractional and Negative Exponents
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Related Solutions
A negative base is a point of conflict between the three commonly used meanings of exponentiation.
- For the continuous real exponentiation operator, you're not allowed to have a negative base.
- For the discrete real exponentiation operator, we allow fractional exponents with odd denominators, and $$(-a)^{b/c} = \sqrt[c]{(-a)^b}= \left( \sqrt[c]{-a} \right)^b = (-1)^b a^{b/c} $$ (and this is allowed because every real number has a unique $c$-th root)
- For the complex exponentiation operator, exponentiation is multivalued. An exponentiation with denominator $n$ generally takes on $n$ distinct values, although one is generally chosen as the "principal" value.
For $(-5)^{2/3}$, these three exponentiation operators give
- Undefined
- $\sqrt[3]{25}$
- $\omega \sqrt[3]{25}$ is the principal value. The other two are $\sqrt[3]{25}$ and $\omega^2 \sqrt[3]{25}$, where $\omega = -\frac{1}{2} + \mathbf{i} \frac{\sqrt{3}}{2}$ is a cube root of $1$.
Unfortunately, which meaning of exponentiation is meant is rarely ever stated explicitly, and has to be guessed from context.
I'm guessing that the second one is meant.
In case you're curious, here is part of the rationale for the first and third conventions.
In the first convention, 'continuity' is important. If two exponents are 'near' each other, then they should produce 'nearby' values when used to exponentiate. However, despite the fact $2/3$, $3/5$, and $\pi/5$ are all similarish in size, $(-5)^{2/3}$ and $(-5)^{3/5}$ are widely separated by the fact one 'should' be positive and the other negative. And it's not even clear that $(-5)^{\pi/5}$ should be meaningful!
For the third convention, the whole thing is like the idea of $\pm 2$ being the 'square root of 4', but for the fact the complexes cannot be cleanly separated into "negative" and "positive" to let us choose a specific one nicely.
A method is chosen for the principal value, based trying to get positive bases 'right' and trying to keep continuity as much as possible, but alas this convention gets the negative bases 'wrong'.
In some sense, this can be viewed as the principal value of $(-5)^{2/3}$ chosen to be "two-thirds of the way" from positive to negative.
There are two reasons. First, chained exponentiation follows the rule:$$ \left(x^y\right)^z = x^{yz}.$$ So, if $y= 1/z$ then $\left(x^y\right)^z = x$, meaning that exponentiation by the reciprocal obeys the property that defines taking a root.
Second, if you want to know how these things are actually calculated, that's an interesting question, too. We can define two functions $\operatorname{e}^x$ and it's functional inverse $\ln x$ (that is $\operatorname{e}^{\ln x} = \ln [\operatorname{e}^x] = x$) in terms of power series. $\ln x$, in particular, has the property: $$\ln x^y = y\ln x,$$ making it possible to calculate $x^y$ using the identity: $$x^y = \operatorname{e}^{y\ln x}.$$ I don't think that this is how it is actually calculated by computers, because this way is probably pretty slow, but it can be done this way.
Best Answer
Conceptually, it's difficult to provide a clear high-level intuition for these things (i.e. an explanation that makes it easier for the students to understand, not more complicated). As these concepts are reasonably straight-forward, you may find it easiest to just teach them these mnemonic devices:
$$x^{-a} = \frac{1}{x^{a}}$$
$$x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$$
When we talk about square roots, it may make it easier to be more consistent and to always write them as $\sqrt[2]{x}$, so that $x^{\frac{1}{2}} = \sqrt[2]{x}$ will make it easier to remember that $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$.