# Tag Info

Accepted

• 9,639

### 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 ...
• 5,629

### 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 ...
• 507

### 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 ...
• 2,656

### 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 ...
• 271

### 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 ...
• 29.9k

### 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 ...
• 237

### 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 ...
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. ...
• 111
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 ...
• 1,206

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