Questions tagged [exponentiation]

Questions about exponentiation, the operation of raising a base $b$ to an exponent $a$ to give $b^a$.

Exponentiation is a mathematical operation which produces a power $a^n$ from a base $a$ and an exponent $n$. The objects involved are usually numbers, but the procedure can be generalized to matrices, elements in algebraic structures, sets, etc.

3996 questions
70
votes
5 answers

How do you compute negative numbers to fractional powers?

My teachers have gone over rules for dealing with fractional exponents. I was just wondering how someone would compute say: $$(-5)^{2/3}$$ I have tried a couple ways to simplify this and I am not sure if the number stays negative or turns into a…
Kot
  • 2,953
  • 15
  • 42
  • 56
69
votes
13 answers

What Is Exponentiation?

Is there an intuitive definition of exponentiation? In elementary school, we learned that $$ a^b = a \cdot a \cdot a \cdot a \cdots (b\ \textrm{ times}) $$ where $b$ is an integer. Then later on this was expanded to include rational exponents, so…
baum
  • 1,369
  • 12
  • 17
66
votes
18 answers

Is there a simple function that generates the series; $1,1,2,1,1,2,1,1,2...$ or $-1,-1,1,-1,-1,1...$

I'm thinking about this question in the sense that we often have a term $(-1)^n$ for an integer $n$, so that we get a sequence $1,-1,1,-1...$ but I'm trying to find an expression that only gives every 3rd term as positive, thus it would…
64
votes
1 answer

Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

I've known that one can arrange all the numbers from $1$ to $\color{red}{15}$ in a row such that the sum of every two adjacent numbers is a perfect square. $$8,1,15,10,6,3,13,12,4,5,11,14,2,7,9$$ Also, a few days ago, a friend of mine taught me that…
mathlove
  • 122,146
  • 9
  • 107
  • 263
62
votes
17 answers

Which is larger? $20!$ or $2^{40}$?

Someone asked me this question, and it bothers the hell out of me that I can't prove either way. I've sort of come to the conclusion that 20! must be larger, because it has 36 prime factors, some of which are significantly larger than 2, whereas…
Alec
  • 3,967
  • 1
  • 24
  • 60
60
votes
7 answers

How can I intuitively understand complex exponents?

Could you help me fill in a gap in my understanding of maths? As long as the exponent is rational, I can decompose it into something that always works with primitives I know. Say, I see a power: $x^{-{a\over b}}$ for natural a,b. Using the basic…
SF.
  • 1,078
  • 8
  • 16
60
votes
7 answers

Finding $\int x^xdx$

I'm trying to find $\int x^x \, dx$, but the only thing I know how to do is this: Let $u=x^x$. $$\begin{align} \int x^x \, dx&=\int u \, du\\[6pt] &=\frac{u^2}{2}\\[6pt] &=\dfrac{\left(x^x\right)^2}{2}\\[6pt] &=\frac{x^{2x}}{2} \end{align}$$ But…
Garmen1778
  • 2,168
  • 3
  • 19
  • 31
56
votes
12 answers

What are the Laws of Rational Exponents?

On Math SE, I've seen several questions which relate to the following. By abusing the laws of exponents for rational exponents, one can come up with any number of apparent paradoxes, in which a number seems to be shown as equal to its opposite…
51
votes
6 answers

How to calculate a decimal power of a number

I wish to calculate a power like $$2.14 ^ {2.14}$$ When I ask my calculator to do it, I just get an answer, but I want to see the calculation. So my question is, how to calculate this with a pen, paper and a bunch of brains.
Mixxiphoid
  • 922
  • 3
  • 13
  • 25
50
votes
13 answers

What exactly IS a square root?

It's come to my attention that I don't actually understand what a square root really is (the operation). The only way I know of to take square roots (or nth root, for that matter) it to know the answer! Obviously square root can be rewritten as…
nuggetbram
  • 581
  • 1
  • 4
  • 7
50
votes
7 answers

Inequality from Chapter 5 of the book *How to Think Like a Mathematician*

This is from the book How to think like a Mathematician, How can I prove the inequality $$\sqrt[\large 7]{7!} < \sqrt[\large 8]{8!}$$ without complicated calculus? I tried and finally obtained just $$\frac 17 \cdot \ln(7!) < \frac 18 \cdot \ln(8!)$$
48
votes
6 answers

What's the inverse operation of exponents?

You know, like addition is the inverse operation of subtraction, vice versa, multiplication is the inverse of division, vice versa , square is the inverse of square root, vice versa. What's the inverse operation of exponents (exponents: 3^5)
warspyking
  • 719
  • 1
  • 8
  • 17
48
votes
6 answers

What is the order when doing $x^{y^z}$ and why?

Does $x^{y^z}$ equal $x^{(y^z)}$? If so, why? Why not simply apply the order of the operation from left to right? Meaning $x^{y^z}$ equals $(x^y)^z$? I always get confused with this and I don't understand the underlying rule. Any help would be…
GambitSquared
  • 3,502
  • 3
  • 30
  • 52
47
votes
4 answers

Is this a new method for finding powers?

Playing with a pencil and paper notebook I noticed the following: $x=1$ $x^3=1$ $x=2$ $x^3=8$ $x=3$ $x^3=27$ $x=4$ $x^3=64$ $64-27 = 37$ $27-8 = 19$ $8-1 = 7$ $19-7=12$ $37-19=18$ $18-12=6$ I noticed a pattern for first 1..10 (in the above…
46
votes
3 answers

What is the value of $1^i$?

What is the value of $1^i$? $\,$
yrudoy
  • 1,893
  • 18
  • 22