Questions about exponentiation
73
votes
8answers
2k views
What does $2^x$ really mean when $x$ is not an integer?
We all know that $2^5$ means $2\times 2\times 2\times 2\times 2 = 32$, but what does $2^\pi$ mean? How is it possible to calculate that without using a calculator? I am really curious about this, so ...
58
votes
5answers
2k views
Root Calculation by Hand
Is it possible to calculate and find the solution of $ \; \large{105^{1/5}} \; $ without using a calculator? Could someone show me how to do that, please?
Well, when I use a Casio scientific ...
40
votes
4answers
1k views
A new imaginary number? $x^c = -x$
Being young, I don't have much experience with imaginary numbers outside of the basic usages of $i$. As I was sitting in my high school math class doing logs, I had an idea of something that would ...
39
votes
4answers
2k views
Are the solutions of $x^{x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}}}=2$ correct?
Problem:
Find $x$ in
$$\large x^{x^{x^{x^{ \cdot^{{\cdot}^{\cdot}} }}}}=2$$
Trick:
$x^{x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}}}=2$, so,
$x^{(x^{x^{x^{\cdot^{{\cdot}^{\cdot}}}}})}=x^2=2$, and,
...
30
votes
11answers
2k views
How is $e^x$ read aloud?
My current research colleague from New Castle told me that I was reading it wrong. I usually read it as e power x.
How do you read aloud $e ^ x$?
Is it:
e raised to x
e power x
e powered x
or e ...
28
votes
5answers
2k views
$x^y = y^x$ for integers $x$ and $y$
We know that $2^4 = 4^2$ and $(-2)^{-4} = (-4)^{-2}$. Is there another pair of integers $x, y$ ($x\neq y$) which satisfies the equality $x^y = y^x$?
24
votes
14answers
2k views
How do I understand $e^i$ which is so common?
Raising something to an imaginary number is weird, I have a hard time wrapping my head around that.
And e seems even more common and comes up in many situations, such as:
the non-geometric ...
23
votes
3answers
1k views
21
votes
1answer
270 views
Iterated exponent of $i$
WolframAlpha seems to tell me that $e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^i}}}}}}}}}} = 1$, see link. Is this just an error or is it for real? Adding one more $e$ to the bottom of the tower gives me the ...
18
votes
6answers
1k views
Why do we need to prove $e^{u+v} = e^ue^v$?
In this book I'm using the author seems to feel a need to prove
$e^{u+v} = e^ue^v$
By
$\ln(e^{u+v}) = u + v = \ln(e^u) + \ln(e^v) = \ln(e^u e^v)$
Hence $e^{u+v} = e^u e^v$
But we know from basic ...
18
votes
3answers
2k views
Can you raise a number to an irrational exponent?
The way that I was taught it in 8th grade algebra, a number raised to a fractional exponent, i.e. $a^\frac x y$ is equivalent to the denominatorth root of the number raised to the numerator, i.e. ...
17
votes
6answers
1k views
$5^{x}+2^{y}=2^{x}+5^{y} =\frac{7}{10}$ Work out the values of $\frac{1}{x+y}$
$5^{x}+2^{y}=2^{x}+5^{y} =\frac{7}{10}$
Work out the values of $\frac{1}{x+y}$
15
votes
6answers
795 views
Why is $x^0 = 1$ except when $x = 0$?
Why is any number (other than zero) to the power of zero equal to one? Please include in your answer an explanation of why $0^0$ should be undefined.
15
votes
4answers
216 views
Intuition for $\omega^\omega$
I'm trying to understand the ordinal number $\omega^\omega$ and I'm having a hard time. I think I understand what $\omega^2$ is. It's what I would get if I took countably many copies of $\omega$ and ...
14
votes
6answers
577 views
A question comparing $\pi^e$ to $e^\pi$ [duplicate]
I was doing an algebra problem set following a chapter on logarithms and exponentiation, and it presented this "bonus question":
Without using your calculator, determine which is larger: $e^\pi$ ...