# Tag Info

31

I agree with vonbrand that it is important to stress that this is a convention that is used sometimes but not others. But I would add the emphasis that all conventions are local. There are places where it is helpful to adopt this convention but other settings in which it would be a disaster. My preference would be to make sure students understand that the ...

29

The most intuitive reason I know for $0^0 = 1$ comes from interpretation in terms of functions, namely $$\text{There are } |B|^{|A|} \text{ functions } A \to B \text{ for any finite A and B}.$$ Now, there are no functions $\{\spadesuit\} \to \varnothing$, so $0^1 = 0$, but there exists exactly one function $\varnothing \to \varnothing$ (its set of ...

16

In the context of polynomials, $0^0 = 1$ is a convenient convention to be able to write $p(x) = \sum_{0 \le k \le n} a_k x^k$ and have the formula valid for $x = 0$ (perhaps explain that here the $0$ is fixed, while $x$ is variable; so the "indeterminate unless the exact path followed in $x^y$" just doesn't come up).

9

Should we impose that $(a^m)^n=a^{mn}$ only when $a \gt 0$? Maybe you should tell your student that he/she have discovered by himself/herself the proof that the rule $(a^m)^n=a^{mn}$ cannot be true with negative bases and rational exponents, and this is a great achievement. Try to explain that he/she proved that If the said rule is true for negative ...

9

Since there is much choice in Terms of limits as you said, a definition should be reasonable in terms of convenience. If you define the exponential function via the series $e^x:=\sum_{n=0}^{\infty} \frac{x^n}{n!}$, then in $e^0$ you need $0^0=1$. The same is true for the binomial theorem. Edit: Also have a look at http://en.wikipedia.org/wiki/...

9

Rigid criteria for simplification seem to me largely a bad idea if they are not motivated by contextual considerations. The idea that $\sqrt{2}/2$ should be preferred to $1/\sqrt{2}$ struck me as unmotivated when I was a student, and now seems to me problematic to motivate. The situation is different with respect to writing rational numbers or rational ...

8

might i suggest a different approach that I have found very helpful when teaching about exponentiation rather than real world examples? I have found that until students understand why a rule works (i.e. the derivation or something similar), they won't be able to understand how to use it. Instead of teaching the rules to the students, have them expand all ...

8

If it is feasible, you could start by introducing the big product operator ($\prod$) and use it to define (or just write) such things as the factorial, the exponentiation of natural numbers or binomial coefficients: $$n! := \prod\limits_{i=1}^n i;\quad a^n := \prod\limits_{i=1}^n a;\quad \binom{n}{k} := \prod\limits_{i=1}^k \frac{n+1-i}{i}$$ Now, it is ...

7

Obviously the correct mathematical answer is to show how the exponent rules actually work, and when they do not work. So please don't accept this answer. Anyway, the educational answer is to see that student is using the fact that $1^2 = (-1)^2$ in the critical middle step. Show them the corresponding fact for cubes, because it's crazy, and might expand ...

6

When I asked the What are the Laws of Rational Exponents? question on SE Mathematics, I was largely thinking about this context; teaching at the level of high school or early (remedial) college math. While it wasn't the top-voted, my answer there represents my best thinking about the status of this issue in classes at that level. As I wrote: Regarding the ...

6

Others have already given good reasons for the usual convention that $0^0=1$. A while ago, I explained at https://math.stackexchange.com/questions/475337/ why I consider the most common alternative, namely to declare $0^0$ undefined, to be inadequately motivated. Specifically, it serves only to protect calculus students from certain sloppy mistakes, and ...

6

Here's something I used to do in college algebra and precalculus classes, from the mid 1980s to the mid 2000s, which has the additional advantage of being an example in which estimation is used. I think you can adapt this to your case. (Use metric system units if appropriate.) Of course, you'll want to go a lot slower than I do below, which is written for ...

5

You could use combinatorics: How many words with 4 letters of 26 letter alphabet do exist? If you have the answer: How many passphrases of two word with 4 letters of the same alphabet do exist? How many passphrases of a 4 letter word and a 6 letter word do exist? The real-world problem arises from the security of passwords (for Facebook for example). ...

5

Answers of @dtldarek and @vonbrand are excellent, but Ī’m ready to add my few cents. Consider k random independent events of zero probability each (for example, Ī shoot for the Moon). Which probability has “all successes” event? Of course, 0k. We have no problems to realize that for k = 1, 2, … “all successes” is a zero-probability event. But what for k = ...

4

Raising a number to a natural-number power, and raising a number to a real-number power, are different operations whose results mostly coincide. In any abstract algebraic ring, raising any element $x$ to a natural number power $n$ is equivalent to multiplying the multiplicative identity of that ring by $x$, $n$ times. If $n$ is zero, the result in any ring ...

3

I would say the canonical answer for what constitutes 'simplified as much as possible' is whatever the exam board says it is. 'Simplify' isn't a mathematical function. It is a pedagogical instruction trying to require students to make use of a selection of mathematical equivalences that they are expected to know. However, it is too vague a term to make ...

3

It costs $5/month (for educators) to use Wolfram Alpha in its practice worksheets model. It will generate a lot of problems for you, but I'm not 100% sure it gives you the granularity you want. I really like it. https://www.wolframalpha.com/pro-for-educators/ Also, Math.com has a worksheet generator, which allows some specification of fraction use and ... 2 Exponential notation is a shorthand for repeated multiplication in the same way that multiplication is a shorthand for repeated addition. Where multiplication returns an addend, exponentiation returns a multiplicand. In multiplication,$0x = 0$since adding no '$x$'s is the same as doing no addition, which is the same as adding zero. In exponentiation,$x^0 ...

2

There is a theorem which says that it is impossible to decide the equivalence of two elementary functions syntactically. So there is not, and cannot, be a uniquely defined "simplest form" for a given expression. http://inst.cs.berkeley.edu/~cs282/sp02/readings/caviness.pdf

1

Something like this? $$a. \left({a+b \over a-b} + {a-b \over a+b}\right) \div \left({a^2 \over a^2-b^2} + {1 \over {a^2 \over b^2}-1}\right)$$ $$b. \left({x^2y - xy^2 \over x-y} + xy\right) \times \left({y \over x} + {x \over y}\right)$$ $$c. \left({n \over m-n} + {m \over {m + n}} \right) \times \left({m^2 \over n^2} + {n^2 \over m^2} - 2\right)$$ d. \...

1

I let the students discover the rules for exponentiation as a consequence of finding prime factors. Students like the shorthand notation $24=2\times2\times2\times3=2^3\times3$ and related questions arise naturally. Given the prime factors of 24 what is the prime factor decomposition of 240? I like to juxtapose $2+2+2+2=4\times2$ with $2\times2\times2\times2=... 1 If working with high school students, I would suggest mentioning that all of the usual exponent rules hold. It could be a fun exercise to ask your students to interpret the numerical value of$0^0\$ and to write a brief paragraph explaining their reasoning.

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