# Tagged Questions

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

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### Does the complex modulus satisfy the power identity $|z^r|= |z|^r$?

Can we "split the modulus" of complex numbers? Let $z\in\mathbb{C}$. Then, does $$|z^{r}|=|z|^r$$ hold, where $r\in\mathbb{R}$. Is this true even for $r\in\mathbb C$ ? Also, can we show this? I am ...
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### Proof: Raising a complex number to a rational power

The problem from the textbook is: Prove that if (a complex number) $z$ is a number on the unit circle, then $z$ has finitely many distinct powers $z^n$ if and only if the argument of $z$ is a ...
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### Question concerning comparison of different tetration functions

Let $a_{1}=2$, $a_{n+1}=2^{a_{n}}$ for $n \geq 1$ Let $b_{1}=3$, $b_{n+1}=3^{b_{n}}$ for $n \geq 1$ Is is true that $a_{n+2}>b_{n}$ for all $n \geq 1$? If so, is the proof elementary? (Use only ...
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### What is the last digit of $7^{2015}$? [closed]

What is the last digit of $7^{2015}$?
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### How can I compare the numbers $2^{39}$, $5^{19}$ and $52^7$?

I have to compare the numbers $2^{39}$, $5^{19}$ and $52^7$. I don't know how to do that because their exponents don't have anything in common.
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### How to represent a $5$ digit number that has $62$ choices per digit?

If you have a $5$ digit number that can be 0-9A-Za-z how would you represent that? total_number_of_records = 5 digits * (10 + 26 + 26) ^ 5 I want to find out ...
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### Infinite tetration of $i$

Proof Euler's identity; $$e^{i\pi} + 1 = 0$$ can be manipulated in order to obtain the result: $$e^{i\pi} = -1$$ Raising both sides of the equality to the power of $i$ gives, after simplification:...
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### Exponentiating roots of unity

When given some number some complex $2n^{th}$ root of unity, $z$, how does one evaluate something such as $z^{2m}$ (with $m=kn$)? I would take $z^{2m}=(z^{2n})^k=1^k=1$, but I don't know if this is ...
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### Unable to process Large numbers [closed]

A small spherical cell of diameter $1.616E^{-35}$ is exponentially multiplying as $2^n$ where n is the generation number. The duration of 1 generation is $5.39E^{-44}$ second. And the cells cluster ...
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### How to calculate 10^ decimal power without a calculator?

I need to know how to calculate 10^ a decimal power, like 10^-7.4, without a calculator, in as simple a way as possible, since I will be doing questions which only allow me about a minute to a minute ...
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### -1 raised to fractional indices lying between 0 and 1

For a personal project, I had to figure out what happens when $-1$ (negative one) is raised to fractional powers lying between $0$ and $1$. I thought that if I get a power $x = 0.a_1a_2a_3...a_n$ (...
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### What is the right way to calculate a power?

I noticed that there are two solutions for $(-1)^{14/2}$: $((-1)^{14})^{1/2} = 1$ $(-1)^{14/2}=(-1)^7=-1$ What am I doing wrong?
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### What exactly are those “two irrational numbers” $x$ and $y$ such that $x^y$ is rational? [duplicate]

It's possible to prove nonconstructively that there exists irrational numbers $x$ and $y$ such that $x^y$ is rational, but that proof only proves that such numbers exist and does not specify what they ...
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### Got stuck while integrating $\int x^x dx$ [duplicate]

What is the integration of $$\int x^x dx$$ And how can I understand whether an integration is possible or not? Is there any rule to understand whether a function is integrable or not?
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### SAT question about integers and exponents

If $a$ and $b$ are positive integers and $$(a^\frac{1}{2}\times b^\frac{1}{3})^6=432$$ What is the value of $ab$?
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### Finding the function that would describe this:

I'm not going to go into detail why I am interested in the next iteration of these functions, but here they are: 1: 6/(x+1) 2: 8/(2^x) 3: 10/(?) The question is, which one is next? I will say that ...
I am currently writing a scriptum struggling with the definition of matrix power. Precisely, let $\mathbb A \in \mathbb C^{n, n}$ and $p \in \mathbb R$. I Currently have: If $p \in \mathbb N$ ...