# How to explain Fractional and Negative Exponents

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?

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We want to define fractional and negative exponents so that the rules $x^m x^n = x^{m+n}$ and $(x^m)^n = x^{mn}$ remain true, even when $m$ and $n$ aren't positive integers. This forces us to use the standard definitions. –  littleO Oct 10 '13 at 7:28
For example, we would hope that $5^0 \times 5^1 = 5^{0+1} = 5$. This means that $5^0$ should be equal to $1$. Also, we would hope that $5^{\frac12} \times 5^{\frac12} = 5^{\frac12 + \frac12} = 5^1 = 5$. This means that $5^{\frac12}$ should be equal to $\sqrt{5}$. –  littleO Oct 11 '13 at 2:04

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 $x^{\frac{m}{n}} = \sqrt[n]{x^{m}}$.

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Negative exponents are pretty easy. Start with the concepts that $a^xa^y=a^{x+y}$ and $a^0 = 1$, and negative exponents are done.

Fractional exponents really come from a couple different directions. One approach is to look at rational ones. Since Newb's answer already describes them in terms of combining powers with roots, I'll give another way: $$(a^x)^y = a^{xy}.$$

Another way is to look at the power series for the exponential function. Yet another is to look at logarithms.

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From OP's profile, I glean that he's trying to teach his 13-year-old classmates. Using power series or logarithms to try to explain fractional exponents may be inappropriate. –  Newb Oct 10 '13 at 7:10
@Newb: I expanded to add another that's likely to be more useful. –  dfeuer Oct 10 '13 at 7:23

Fractional exponents might be explained in the usual ways that fractions are. For example one supposes that there is a $y^a = x$, and thus $x^{1/a}=y$. You could even go down as far as semitones etc, or some diagrams like this post at the dozenal society, which deal with fractional logs, and how we can severely limit the ratios of logs 2, 3, 5 by nothing more than tests along $x<y<100$, for example. It continues to the next message.

http://z13.invisionfree.com/DozensOnline/index.php?showtopic=718&view=findpost&p=22047479

Negative logrithms are like negative numbers, but one has to be sure that the bar-notation used in pre-calculator days, represents eg $-1+0.30103$, rather than $-0.69897$ for $\log_{10} 0.2$.

Most of what you can do with fractions, you can do with exponents.

Look at http://z13.invisionfree.com/DozensOnline/index.php?showtopic=1025&view=findpost&p=22099409 where a table of $10^{a\times 10^{-b}}$ is given. I used such a table to find logrithms way back in the 1970s on a four-function calculator. It was more accurate and faster than the seven-figure logrithm tables.

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Knowing is completely different from understanding. You and your classmates are asking the exactly right questions.

After we adopt the notation for raising a number to natural (positive integer) power to stand for repeated multiplication by itself, the question naturally arises of whether it would also be meaningful for the power to be zero, or a fraction, or negative (or, later, imaginary or complex etc.). All mathematics is developed this way.

A brilliant idea is to suggest that if the PROPERTIES of raising a number to a natural power are satisfied, we accept accept the result as meaningful. That is, if a^b a^c = a^(b+c) and (a^b)^c = a^(bc), then we accept the newcomer. this is how we determine that a^0, (a≠0) =1. So we ask if a^-b times a^b equals a^0 which we know is 1. The answer is yes. That's it. This is how we arrive at our conclusions. We use the known properties to answer the question. In other words, we require consistency. It's a bit more complicated to determine the meaning of 0^0 but the basic idea is the same. (It too turns out to be 1.)

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