# Tagged Questions

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

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### number raised to a logarithmic fraction

I know and saw that $n^\frac{1}{\lg n} = 2$ but I don't understand how this can be. What am I missing?
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8k views

### Calculating power without using a calculator, for example $1.05^{10}$

How to find (or estimate) $1.05^{10}$ without using a calculator? Do we have any fast algorithm for cases where base is slightly more than one? Say up to $1.1$ with tick $0.05$.
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### How to differentiate x^(1/x)?

How to differentiate the following? $$x^{\frac{1}{x}}$$ (I know the answer is $\frac{1-\ln(x)}{x^{2-\frac{1}{x}}}$, but I do not understand how to get there) Attempt at solution I believe the ...
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### Solve $2^x+2^{-x} = 2$

Need to solve: $$2^x+2^{-x} = 2$$ I can't use substitution in this case. Which is the best approach? Event in this form I do not have any clue: $$2^x+\frac{1}{2^x} = 2$$
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### Calculate, simplify and expand exponents with complex numbers

Can we somehow calculate $a^z$ where z is a complex number ? Does normal exponent rules like : $$a^b\cdot a^c=a^{b+c}$$ Still work when complex numbers are in the exponent ? For example, do these ...
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### What operations have represented 2^2? [closed]

I knew = 2.2 and 2.2= 2+2 What operations have represented 2^2? ? = 3+3+3=9 ; 3.3.3=81 and continue is . what is representation for operations ? I assume to have a operation, it is called f. I ...
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### Why don't parentheses matter in this case of multipication

Very basic question but can't seem to wrap my head around why this happens. Normally parentheses indicate that the operation inside must be carried out first. In this case: (a * a * a)*(a * a * a * ...
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### Exponents using non integers

Let’s imagine that the number of rabbits in a field doubles each month. If we start with 6 rabbits, how many rabbits would be in the field after 10 weeks, given that 4 weeks = 1 month? I would think ...
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### How to evaluate $\log(1 - x)$ in terms of $\log(x)$?

I can do this using the following relation: $$\log(1 - x) = \log(1 - \exp(y))$$ Here $y = \log(x)$ is always a negative number. However, I was wondering whether it's possible to compute $\log(1 - x)$...
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### Taking the square root of an imaginary number

We know that when we take the square root of a negative real number, it's realness "splits open" and an "imaginary" dimension is introduced (characterized by the presence of iota). The question is, ...
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### Finding every triplet $(n,a,b)$ such that $n!=2^a-2^b$

Question : Let $n,a,b$ be positive integers. Are there infinitely many triplets $(n,a,b)$ which satisfy the following equality?$$n!=2^a-2^b$$ If Yes, then how can we prove that? If No, then how can ...
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### Can 720! be written as the difference of two positive integer powers of 3?

Does the equation: $$3^x-3^y=720!$$ have any positive integer solution?
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### Exponentiation with negative base and properties

I was working on some exponentiation, mostly with rational bases and exponents. And I stuck with something looks so simple: $(-2)^{\frac{1}{2}}$ I know this must be $\sqrt{-2}$, therfore must be ...
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### Find the $n$-th power of complex number

Let $z=1+2i$ be a complex number. Prove that for any $n \in \mathbb{N}^*$, the number $z^n$ has the following form: $a_n+ib_n$, with $a_n,b_n \in \mathbb{Z}$. I guess the solution lies in the ...
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### Jordan form of a power of Jordan block?

How, in general, does one find the Jordan form of a power of a Jordan block? Because of the comments on this question I think there is a simple answer.
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### How is Faulhaber's formula derived?

I have been wanting to understand how to find the sum of this series. $$1^p + 2^p + 3^p +{\dots} + n^p$$ I am familiar with Gauss' diagonalised adding trick for the sum of the first $n$ natural ...
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### matrix exponential limit

I'm having litlle trouble here to prove the following statement: "Let $A$ an $n\times n$ matrix (real or complex). Prove that $$\lim_{n \to \infty} \left(I + \frac{A}{n}\right)^{n} = e^{A}.$$ Now I'...
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### Why is X raised to the power of 0 = 1?

So as the topic states, why is 5^0 = 1 and not 5 or 0? Is it only because the other exponential laws wouldn't work if it was not the case?
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### Powers in ASCII text

I have a problem reading a discussion forum post. Namely, in the ASCII text, is 2^3^4 the same as $(2^3)^4$ or $2^{3^4}$?
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### 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 ...
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### How to solve $1/{4a^{-2}}$

Can i write $1/4a^{-2}$ as $4a^2$ ? Or is the right answer to do it like: $$1/4a^{-2} = 1/4 \cdot 1/a^{-2} = 1/4 \cdot a^2 = a^2/4$$ In the problem there is no parenthesis around $4a$ but assuming ...
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### What is the actual meaning of power which is less than 1? [duplicate]

$3^4 = 3 \cdot 3 \cdot 3 \cdot 3$ $3^2 = 3 \cdot 3$ $3^1 = 3$ But my problem is for below 1: Example: $3^{0.7} = \ ?$ Can you please explain it to clear my doubt?
### Function $f: \mathbb{Z} \to \mathbb{Z}^n$ related to $\sum_{k=1}^{x} k^n$.
The sequence $\{a_0,a_1,...a_x\}$ has closed form $a_n=\sum_{i=0}^{\infty} \Delta^i(0) {n \choose i}$ where $\Delta a_n$ denotes the operation mapping $a_n$ to $a_{n+1}-a_n$ and $\Delta^i(0)$ is ...