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27 votes
Accepted

Natural occurrences of a to the (b to the c)?

If your students have learned some statistics, then you could point out that the normal distribution's probability density function uses this double exponential. $$f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\...
JRN's user avatar
  • 10.8k
15 votes

How should I convince a student who thinks they proved $1=-1$

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 ...
Pedro's user avatar
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12 votes

Natural occurrences of a to the (b to the c)?

Count all possible functions mapping $i$ input bits to $o$ output bits: For each of the $2^i$ input combinations, each output has two possible outputs ($0$ and $1$), i.e. you have $2^{2^i}$, leading ...
Tobias Kienzler's user avatar
9 votes

How should I convince a student who thinks they proved $1=-1$

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 ...
Chris Cunningham's user avatar
8 votes

Natural occurrences of a to the (b to the c)?

The number of undirected graphs of order $n$ is $2^{n \choose 2} = 2^{(n^2-n)/2}$ (e.g. consider the adjacency vector as a binary-encoded number). You can describe this in terms of the number of ...
Yonatan N's user avatar
  • 181
7 votes

Rational Powers of Negative Numbers on Basic Calculators

Answer: The calculators are fine, it is the question's premise "$(-8)^{2/3} = 4$" that needs to be fixed. Now this was a bit polemic, of course. Here's a more detailed description of what I ...
Jochen Glueck's user avatar
7 votes

Searching activities with "Find the error" strategy for learning maths

As to question 1: I do have a german speaking background, so please excuse the source actually is in german: There is a really nice journal called Wurzel, the german word for "square root". In there, ...
SCS's user avatar
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6 votes

How should I convince a student who thinks they proved $1=-1$

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. ...
Daniel R. Collins's user avatar
6 votes

Exponents with Negative Base; with or without Parentheses

You might explain that BEDMAS is not the whole story when it comes to the order of operations. There is an operation called negation. It reverses the sign on numerical quantifies. It gives the ...
Dan Christensen's user avatar
6 votes

Exponents with Negative Base; with or without Parentheses

I like the presentation on the NCTM Math Forum/Dr. Math website: We don't usually list unary operators in PEMDAS because they're thought of as being implied by the rules for binary operations. You ...
Daniel R. Collins's user avatar
5 votes

Exponents with Negative Base; with or without Parentheses

I tutored a student who had a hard time understanding this, and the way that helped him to understand it was this: Any time there is a negative sign on a number, we can read it as $(-1)$ . So $-5 \...
Davy M's user avatar
  • 150
4 votes

Exponents with Negative Base; with or without Parentheses

If your students already understand that exponents precede multiplication, and that multiplying by $-1$ is the "negation" operator, then you should be able to convince them that $$-5^2 = -1*5^2 = -1*...
Nick C's user avatar
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4 votes

Natural occurrences of a to the (b to the c)?

If $S$ represents the set of $n$ students of a school, then $\mathscr P(S)$ is the set of rosters for all possible clubs that that school could host (if we assume that any two clubs with exactly the ...
Matthew Daly's user avatar
  • 5,619
4 votes

Natural occurrences of a to the (b to the c)?

Gompertz model was created in 1825 to study human mortality curves. From the 1920's, it was used in economic fields, and from there it was also used in Biology to study cells and microorganisms, such ...
FormerMath's user avatar
3 votes

Natural occurrences of a to the (b to the c)?

It is easy to square a number. So instead of computing $\exp(z)$ you can compute $\exp(z/2^k)^{2^k}$ for some suitably large positive integer $k$. This is a very simple way of accelerating the ...
user21820's user avatar
  • 2,649
3 votes

Natural occurrences of a to the (b to the c)?

The maximum of a large number of independent, identically distributed random variables -- e.g., the highest flood observed over a long period -- has an extreme value distribution. One common case is ...
nanoman's user avatar
  • 271
3 votes

Natural occurrences of a to the (b to the c)?

What a nice collection of answers! I am inclined to use $\lfloor A^{3^n} \rfloor$ where $A$ is Mill's constant, $$A \approx 1.3063778838630806904686144926 \;, $$ just because it is astounding that ...
Joseph O'Rourke's user avatar
2 votes

Natural occurrences of a to the (b to the c)?

Inspired by your example, I like the number of matrices, tensors, binary or n-ary relations over a specified set (where the domain and range of such a relation need not match), such as asking about ...
Vandermonde's user avatar
2 votes

How should I convince a student who thinks they proved $1=-1$

I think the previous answers have focused too much on the details of rational exponents and negative bases. There is a much simpler point about logic that resolves this whole example and that I think ...
kjfhglksdh's user avatar
1 vote

Natural occurrences of a to the (b to the c)?

Let n be given and consider the set of logical formulas with $n$ variables. Two formulas are equivalent if they are both true, or both false, for any assignment of truth values to the variables. ...
Mark Dominus's user avatar
1 vote

How should I convince a student who thinks they proved $1=-1$

(My new and hopefully improved answer) Should we require that $(a^m)^n=a^{mn}$ only when $a \gt 0$ ? That might "solve" the problem in some sense, but we do have legitimate cases with negative ...
Dan Christensen's user avatar

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