I'm looking for a way so that I can get big numbers.  I want to calculate 38^n for n>4000   So is there any way to do this?  Kindly help me.
What is the easiest way to handle huge numbers in C? I need to store values in the Area 1000^900, or in more human readable form 10^2700. Does anybody know of an easy way to do that? Any help would really be appreciated!
Let $V=\{f : S \to \mathbb{R} \}$ be the set of all the functions $f : S \to \mathbb{R}$. Let $S=\{ 1,2,...,n$}, now my train of thought was that the basis of V must be a linearly independent set that Span V so somehow show that there is a set of functions on S that can compose as a linear combination and map to any number in $\mathbb{R}$. Any idea how to get a general answer for basis for dimension of V? Python metaheuristic packages integral roots for $f(x) = 41$ if $f(x) = 37$ has 5 distinct integral roots. Is the money you get from shorting a stock free to use for going long on other stocks? I shorted a particular stock which resulted in $10,000 being deposited into my Interactive Brokers account. I then put all of that $10,000 into VTI (total stock market index). Basically making the bet that the stock I shorted will perform worse than the total stock market.   I assumed I would be charged some sort of interest or fees for using that money, however it appears that I am not being charged at all. Are there really no costs for doing this? The covid19 death data for US shows strong periodic trends. It does not make sense to me that people die periodically.   What could be the reason to having such data? I thought it might be the reporting artifacts, for example, people always update data at Monday, but the period seems to be 3-4 days. Are if else statement equivalent to logical and && or || and where should I prefer one over the other? I'm learning about decision making structures and I came across these codes:  if [ -f ./myfile ] then      cat ./myfile else      cat /home/user/myfile fi   [ -f ./myfile ] &amp;&amp; cat ./myfile || cat /home/user/myfile   Both of them behave the same. Are there any advantages to using one way from the other? PWM complementary square output waveforms PWM channels in certain microcontrollers such as the SAM3X8E have two complementary square output waveforms, PWMH and PWML, per channel. What is the purpose of having two waves that are the inverses of each other? How to break the Y axis in an Excel sheet I want to break the Y-axis in an Excel sheet. How can I do that?  For example, suppose I have the data in a range from 0-7000 and then range from 22000-28000. I want a break in the Y axis from 8000-20000. What was the point of the stormtroopers' armour? It seems that it didn't actually protect them from any form of attack.   Every direct hit from a laser seemed to kill a stormtrooper outright, and a slice from a lightsaber cuts straight through. Which is correct: "drive safe" or "drive safely"? How do I split a string so I can access item x? Using SQL Server, how do I split a string so I can access item x?  Take a string "Hello John Smith". How can I split the string by space and access the item at index 1 which should return "John"? Words to describe people with good/poor vocabularies Words to describe people with good/poor vocabularies. I am looking for words which I can use to describe characters and I would also like to describe their vocabularies, and explain how their vocabularies stem from their upbringing How many ways can N elements be partitioned into subsets of size K? In grade school we learn to rationalize denominators of fractions when possible. We are taught that $\frac{\sqrt{2}}{2}$ is simpler than $\frac{1}{\sqrt{2}}$.  says that "there is a bias against roots in the denominator of a fraction". But such fractions are well-defined and I'm failing to see anything wrong with $\frac{1}{\sqrt{2}}$ - in fact, IMO it is simpler than $\frac{\sqrt{2}}{2}$ because 1 is simpler than 2 (or similarly, because the former can trivially be rewritten without a fraction).  So why does this bias against roots in the denominator exist and what is its justification? The only reason I can think of is that the bias is a relic of a time before the reals were understood well enough for mathematicians to be comfortable dividing by irrationals, but I have been unable to find a source to corroborate or contradict this guess. Which is correct if I want to request for a pen?        Can I have your pen please?   May I have your pen please? How to use articles with scientific words, such as mass, force, impulse, field and others? How are articles used with scientific quantities, such as mass, force, impulse, field and others? Maximum power transfer proof I have the following homework problem.  Consider a power supply with fixed emf $ε$ and internal resistance $r$ causing current in a load resistance $R$. In this problem, $R$ is fixed and $r$ is a variable. The efficiency is defined as the energy delivered to the load divided by the energy delivered by the emf.  When the internal resistance is adjusted for maximum power transfer, what is the efficiency?  The answer is 50% but I'm confused how this is calculated. Here is my thought process so far.  The power dissipated in the load is $I^2R$ and $\displaystyle I = \frac{ε}{r+R}$ so $\displaystyle P = \frac{ε^2R}{(r+R)^2} = \frac{ε^2}{\frac{r^2}{R} + 2r + R}$ and so power transferred to the load will be maximum when $\displaystyle \frac{r^2}{R} + 2r + R$ is a minimum. Taking the first derivative $\displaystyle \frac {d}{dr}(\frac {r^2}{R} + 2r + R) = (\frac{2r}{R} + 2)$ and this equals 0 when r = -R. The second derivative is $\displaystyle \frac{2}{R} &gt; 0$ so this point would be a minimum and therefore max power transferred when r = -R.  But having $r = -R$ doesn't make sense to me. From what I read, for maximum power transfer $r$ should equal $+R$. So I've probably done something wrong in my calculations or assumptions in the previous paragraph. Also wouldn't max power be transferred when the internal resistance is as close to zero as possible?  Can someone please show me a proof that shows why $r = R$ for max power transfer and then how to calculate the efficiency. Just downloaded a .zip file from the internet. I want to use the terminal to unzip the file. What is the correct way to do this? Why no ooVoo on ubuntu? The only reason I ever boot into Windows (7) is to use the popular video chat software ooVoo (). It's useless under Wine and there seems to be no way to run it in Ubuntu except for some proprietary tools. Does anyone have any information on some alternatives to running it with Wine? How to prove for each positive integer $n$, the sum of the first $n$ odd positive integers is $n^2$? I'm new to induction so please bear with me. How can I prove using induction that, for each positive integer $n$, the sum of the first $n$ odd positive integers is $n^2$?  I think $9$ can be an example since the sum of the first $9$ positive odd numbers is $1,3,5,7,9,11,13,15,17 = 81 = 9^2$, but where do I go from here.