# Tagged Questions

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

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### Is there an algorithm to split a number into the sum of powers of 2?

Am I able to split, lets say 76, into the sum of powers of two, through an algorithm and without cycling through possible combinations? For the example above, the answer would be '2^6+2^3+2^2' or ...
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### sample size calculation for count data

I have a plan to see some treatment effect in several projects in my company where I will compare the average number of errors now and after the treatment. So what I know from current situation is ...
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### 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 ...
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### Supremum and infimum of powers with rational exponents. [duplicate]

I asked this question earlier and got a very useful hint, but I still can't get the full solution. Here is the original question: I'm trying to show that for $b>1$, $x>0$ and $x$ irrational, ...
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### An interesting log problem $3^{{(log_3{x})}^2}+x^{log_3x}=162$

$$3^{{(\log_3{x})}^2}+x^{\log_3x}=162$$ How do I go about doing this. I am stuck at the step $x^{\log_3x} = 81$. Is this right? How do I continue or is it wrong?
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### Trying to show that $\sup \{b^p:p\in Q,\;0<p<x\} = \inf\{b^q:q\in Q,\;x<q\}$

I'm trying to show that for $b>1$, $x>0$ and $x$ irrational, that $$\sup \{b^p:p\in Q,\;0<p<x\} = \inf\{b^q:q\in Q,\;x<q\}$$ I know this follows immediately if we define ...
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### If $a^{(b^c)}=d^c$, find $d$ in terms of $a$ and $b$.

Is it possible to express $d$ in terms of $a$ and $b$ only in the following equation? $$a^{b^c}=a^{(b^c)}=d^c$$ I want something like $d=\dots$ Thanks in advance!
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### Exponential Factorial vs Tetration

I'm wondering whether there's a known way to compare the exponential factorial of n versus the tetration of a fixed number $($ e.g., $3$, since it appears in Graham's number $)$ with the same number ...
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### What is the name of the answer to exponentiation?

What is the name of the answer to exponentiation? Adding two numbers produces a sum. Multiplying two numbers produces a product, but I cannot think of or find the name for the solution to ...
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### Solving an Equation Involving Modular Exponation

Is it possible to solve the following equation? $$3^k\mod k \equiv 24$$ Clearly $k>24$; does a solution definitely exist, and can it be found in any simple way? I can certainly calculate ...
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### Find real constants $c$ and $k$ such that $y=cx^k$ passes through point $(a, b)$ with slope $m$

In the Cartesian plane, can a power function of the form $y=cx^k$ (where $c>0$ and $k>1$, not necessarily an integer) be found such that its graph passes through any arbitrary point $(a, b)$ ...
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### Negative exponents on a quantity in scientific notation considering significant figures

Are there rules that apply to negative exponents with regard to scientific notation? The specific problem is: $$\left(6.3\times10^{2}\right)^{-6}$$ I believe the following is correct: ...
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### Number of integer triples to exponential equation

I'm taking a class on number theory and this is one of the problems my professor gave. How many ordered integer triples $(x,y,z)$ are there such that $x^y-y^x=2017\times z$, where $x,y$ are less than ...
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### Convert $3^n$ to some form of $2^n$

I am not from Maths field, but I need your help to convert the $3^n$ form to $2^n$. I need to change the base from $3$ to $2$. The resultant expression can be of any form.
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### Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator

Prove that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator. I did in the following way. Are there other ways? Proof : Let $f(x)=e\pi\frac{\ln x}{x}$. Then, ...
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### Why is $\gcd(2^p + 1,3^p + 1) = 1$?

Let $p$ be an odd prime. Why is $\gcd(2^p + 1,3^p + 1) = 1$ ? I tried using fermat's little and $\gcd(a+b,a) = gcd(a,b)$ but without succes. I can make a statistical argument that suggests there are ...
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### 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 ...
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### How do you prove that for all real numbers c, there always exists a value n, in which $4^n > c3^n$?

What we have so far, its $\frac{4^n}{ 3^n} > c,$ such that $\left(\frac{4}{3}\right) ^ n > c.$ From that I know that it is true, but I don't really know how to prove it formally.
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### What is the value of $1^i$?

What is the value of $1^i$? $\,$
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### Exponentials with three variables: solving for an equation

$2^{x}=5^{y}=100^{z}$ Find $z$ in terms of $y$ and $x$. The term $z$ should be a function of $x$ and $y$, i.e.: $z(x,y)$. All I could get were recursive attempts.
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### How would I calculate sum of digits in the number (a^b)?

I was doing a question from a site,project euler specifically.I came to a question in which I was asked to calculate sum of digits in number 2^1000.Since I program very often I was able to do that ...
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### How to Find the Derivatives of $\tan^2(x^4)$ and $\sec^3(x^5)$?

I am to find the derivative of f(x) and g(x): So far, I know the following: The derivative of tan(x) = sec(x)^2 The derivative of sec(x) = sec(x)tan(x) So, I have tried the following steps to ...
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### Exponentiating a matrix

I was just wondering whether my solution is correct or not and if it isn't, where I went wrong? Find $e^A$ where $A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ So what I ...
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### Negative number squared in expression $-5-8^{2}$

I know this probably is a silly stupid question, but I just don't get it. I'm currently doing Khan Academy pre-algebra and stumbled upon an awkward problem. I assume that: $-5-8^{2}=59$ Because -8 ...