# Questions tagged [exponentiation]

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

3,062 questions
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### math fallacy problem: $-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1$?

I know there is something wrong with this but I don't know where. It's some kind of a math fallacy and it is driving me crazy. Here it is: $$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$
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### 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 ...
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### Is $2^{16} = 65536$ the only power of $2$ that has no digit which is a power of $2$ in base-$10$?

I was watching this video on YouTube where it is told (at 6:26) that $2^{16} = 65536$ has no powers of $2$ in it when represented in base-$10$. Then he - I think as a joke - says "Go on, find another ...
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### 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 ...
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### What is the *middle* digit of $3^{100000}$?

The decimal representation of $3^{100000}$ has $47713$ digits. What is the $23857^{th}$ digit - i.e. the one in the $10^{23856}$'s place? There are lots of questions on this site asking for the last ...
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### 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 ...
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### Underlying Reason For Taking Log Base 10

For the equation $2^x = 7$ The textbook says to use log base ten to solve it like this $\log 2^x = \log 7$. I then re-arrange it so that it reads $x \log 2 = \log 7$ then divide the RHS by $\log 2$ ...
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### 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.
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### What is the reasoning behind this exponents question?

What is $3^{3^{3}}?$ Plugging $3^{3^{3}}$into the calculator gives 7625597484987. I believe because this implies that $3^{3^{3}}=3^{27}$, is this true? And plugging $(3^{3})^{3}$ gives 19683, ...
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### 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 ...
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### 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 ...
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### Is $\exp(x)$ the same as $e^x$?

For homework I have to find the derivative of $\text {exp}(6x^5+4x^3)$ but I am not sure if this is equivalent to $e^{6x^5+4x^3}$ If there is a difference, what do I do to calculate the derivative of ...
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### Rational number to the power of irrational number = irrational number. True?

I suggested the following problem to my friend: prove that there exist irrational numbers $a$ and $b$ such that $a^b$ is rational. The problem seems to have been discussed in this question. Now, his ...
### Generalisation of $(n+3)^2-(n+2)^2-(n+1)^2+n^2=4$
After seeing the neat little identity $(n+3)^2-(n+2)^2-(n+1)^2+n^2=4$ somewhere on MSE, I tried generalising this to higher consecutive powers in the form $\sum_{k=0}^a\epsilon_k(n+k)^p=C$, where $C$ ...
### Which is greater, $98^{99}$ or $99^{98}$? [duplicate]
Which is greater, $98^{99}$ or $99^{98}$? What is the easiest method to do this which can be explained to someone in junior school i.e. without using log tables. I don't think there is an ...