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

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7
votes
5answers
382 views
0
votes
2answers
66 views

How do I evaluate this:$\sum_{n=1}^{\infty}\frac{1}{n²}(e^x −1 −\frac{x}{1!} −\frac{x²}{2!}−\cdots\frac{x^n}{n!})$?

How do i evaluate this sum :$$\sum_{n=1}^{\infty}\frac{1}{n²}(e^x −1 −\frac{x}{1!} −\frac{x²}{2!}−\cdots\frac{x^n}{n!})$$ Note: I 'd surprised if it is convergent Thank you for any help.
6
votes
6answers
5k views

Negative 1 to the power of Infinity

Can anyone explain me what the result of $$\lim_{n\rightarrow\infty} (-1)^n$$ is and the reason?
1
vote
1answer
39 views

Why is this true: $1- (1-1/n)^{\varepsilon n} \leq \varepsilon + \mathcal{O}(\varepsilon^2)$

In my lecture notes, the following is written: $$1- (1-1/n)^{\varepsilon n} \leq \varepsilon + \mathcal{O}(\varepsilon^2)$$ as $\varepsilon \rightarrow 0$ and $n$ some fixed constant (non-negative ...
8
votes
5answers
1k views

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, ...
1
vote
2answers
436 views

How to find out the value of numbers having fractional powers

How to find out the value of numbers having fractional powers manually without using logarithms and calculators?? For example : $2^{1.6}, 3^{2.1}, 5^{3.22}$ etc, I know we can find out the value using ...
0
votes
5answers
63 views

Is there any notation for general $n$-th root $r$ such that $r^n=x$?

As we know that the notation for the $n$-th principal root is $\sqrt[n]{x}$ or $x^{1/n}$. But the principal root is not always the only possible root, e.g. for even $n$ and positive $x$ the principal ...
0
votes
0answers
26 views

Properties of exponentiation proof

I'm trying to prove the following: "Let $x, y$ be non-zero rational numbers, and let $n,m$ be integers. Then we have $x^n x^m = x^{n+m}$." I've managed to prove by induction the case $n,m \geq 0$ ...
5
votes
4answers
2k views

Examples of continuous growth rates greater than exponential

I read on Wikipedia that growth rate of a function can sometimes be greater than exponential. Can you give me some examples of such functions (preferably continuous ones)? Obviously $x^x$ grows ...
0
votes
1answer
43 views

How can I raise a Taylor Series to a power?

I have recently been undertaking the challenge of finding the antiderivative of $x^x$. In doing so, I have come across the idea of raising a Taylor series to a variable exponent. I came to the ...
0
votes
3answers
77 views

Inductive proof for familiar exponent rule $a^n\cdot b^n = (ab)^n$ [on hold]

Using an inductive proof for all integers $n \geq 1$, prove that $a^n \cdot b^n = (ab)^n$. Can't use only algebra and the commutative law.
5
votes
2answers
314 views

Algorithm for tetration to work with floating point numbers

So far, I've figured out an algorithm for tetration that works. However, although the variable a can be floating or integer, unfortunately, the variable ...
0
votes
1answer
32 views

Formula for $\sum_{i = 1}^n k^n$ [duplicate]

I know from my calculator the answer is $\sum_{i = 1}^n k^n$ = $\frac{k^{n+1}-k}{k - 1}$. I'd just like help understanding why.
2
votes
3answers
77 views

Sum of super exponentiation

$f(x,n)=x^{2^{1}}+x^{2^{2}}+x^{2^{3}}+...+x^{2^{n}}$ Example: $f(2,10)$ mod $1000000007$ = $180974681$ Calculate $\sum_{x=2}^{10^{7}} f(x,10^{18})$ mod $1000000007$. We know that $a^{b^{c}}$ mod ...
8
votes
1answer
98 views

For which complex $a, b, c$ does $(a^b)^c=a^{bc}$ hold?

Wolfram Mathematica simplifies $(a^b)^c$ to $a^{bc}$ only for positive real $a, b$ and $c$. See W|A output. I've previously been struggling to understand why does $\dfrac{\log(a^b)}{\log(a)}=b$ and ...
0
votes
2answers
54 views

What determines what base the right side of this base coversion will be?

Referring to this example of positional notation on Wikipedia: There are several examples $$465\;\;\text{(base 10)} = 465\;\;\text{(base 10)}$$ But then $$465\;\;\text{(base 7)} = ...
0
votes
1answer
19 views

Harmonic Mean Solution

The harmonic mean of two positive numbers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x, y)$ with $x < y$ is the harmonic mean ...
1
vote
1answer
113 views

How can one solve $1^x=2$?

Sure, common sense says there's no solution. But, I feel, there should be one! (If there isn't, can't we construct one?)
-5
votes
3answers
62 views

What's the value of $i^i$? [duplicate]

What's the value of $i^i$?Is it real or imaginary?[$i$ here denotes imaginary number.]
0
votes
1answer
42 views

Defining exponentiation on the integers

If one defines the integers as equivalence classes of pairs of natural numbers, there is a (canonical?) way to define addition and multiplication for the integers based on addition and multiplication ...
0
votes
3answers
63 views

Solve $3^{2x} -2 \cdot 3^{x+5} + 3^{10} = 0$ for $x$

Here's the question: Solve for $x$ in $$3^{2x} - 2 \cdot 3^{x+5} + 3^{10} = 0$$ I know that you have to factor something out, I'm just not sure what that something is. Thanks in advance
2
votes
2answers
44 views

What is the logic/theorem/derivation behind finding the exponent of p in n! By [n/p] + [n/p^2] + [n/p^3] + …? [duplicate]

The exponent of prime number of 3 in 100! is 48. It means 100! is divisible by $3^48$ $$E_3(100!) = \left\lfloor\frac{100}3\right\rfloor + \left\lfloor\frac{100}{3^2}\right\rfloor + ...
0
votes
0answers
31 views

Exponentiation of Pascal's Triangle(in matrix form)

I want to find a pattern in subsequent exponentiations of the pascal triangle shown in the form below Matrix P[K+1][K+1]: $$ \begin{matrix} \binom{0}{0} & 0 & 0 & 0\cdots ...
-2
votes
1answer
70 views

Which is the largest power of natural number that can be evaluated by computers? [closed]

Which is the largest power of natural number that can be evaluated by computers? For example if we take a very large power of 7: $7^{120000000000}$. Can a computer calculate this number?
3
votes
5answers
208 views

Solve this logarithmic equation: $2^{2-\ln x}+2^{2+\ln x}=8$

Solve this logarithmic equation: $2^{2-\ln x}+2^{2+\ln x}=8$. I thought to write $$\dfrac{2^2}{2^{\ln(x)}} + 2^2 \cdot 2^{\ln(x)} = 2^3 \implies \dfrac{2^2 + 2^2 \cdot ...
0
votes
3answers
47 views

Find remainder of $\frac{17^{235}}{ 23}$

I need to find remainder of $\frac{17^{235}}{ 23}$. This is supposed to be solved using the following method: $\varphi(23) = 22$ ${17}^{235} = (({17}^{22})^{10})\cdot {17}^{15}$ ${17}^{22}\equiv 1 ...
-4
votes
2answers
76 views

What is the solution of $2^{2^{2^2}}$? [closed]

Which of the following values is same as $2^{2^{2^2}}$ ? $2^6$ $2^8$ $2^{16}$ $2^{222}$ What if it is $a^{b^c}$ ? Is it $a^{(b^c)}$ or $a^{bc}$ ?
155
votes
21answers
12k views

Zero to the zero power - Is $0^0=1$?

Could someone provide me with good explanation of why $0^0 = 1$? My train of thought: $x > 0$ $0^x = 0^{x-0} = 0^x/0^0$, so $0^0 = 0^x/0^x = ?$ Possible answers: $0^0 * 0^x = 1 * 0^x$, so ...
6
votes
6answers
117 views

What are the last two digits of $77^{17}$?

I'm trying to solve current task referenced the following but I stuck at $\displaystyle77^{17}\equiv x\pmod{100}$. As it is described on above link it uses Binomial Theorem. But I read a lot about the ...
5
votes
4answers
73 views

Show $x^n \geq x_1^n+x_2^n+\ldots+x_k^n \Bigg\vert x_1+x_2+\ldots+x_k = x, x \geq 0, n \geq 1, k\in\mathbb{Z}$

Hello StackExchange Community, This is my first post on the forum. Please forgive me for any errors with formatting and my expressions. I am working on the following proof: Show $$x^n \geq ...
2
votes
3answers
91 views

Find all $x$ such that $2^x,2^{x^2}$ and $2^{x^3}$ form $3$ terms of an A.P.

I know that if $a,b,c$ are in Arithmetic Progression, then $2b=a+c$, but in this case, I am unable to solve for $x$. Hints are appreciated. Thanks.
1
vote
2answers
41 views

Help with some simpler symmetric group $S_n$ problems.

I apologize if the problems seem trivial but I have not been able to find example problems or solutions to some of these questions. Could someone please confirm my attempts are correct or not? ...
4
votes
4answers
151 views

How to find the value of $x$ in $x^5=32$

I understand that $2^5=32$ but how would one go about finding it without doing any guessing (what if the numbers were much greater)?
1
vote
1answer
73 views

Homework help to rearrange formula

Given the equation $${V_m} = u(\ln {m_0} - \ln {m_8}) - g{t_f}$$ I need to solve for ${m_0}$ Here is what I have but it looks messy and I feel like there is sometihng wrong or a better way 1st ...
-1
votes
2answers
57 views

Evaluate $2^{-n}(2^n-2^{1+n})$

The answer is $-1$, but how does one expand and simplify this expression to find this answer (what are the steps)?
2
votes
1answer
42 views

Confusion about exponents like ${x^m}^{(1/n)}$.

I've been reading this post. It says that $\sqrt[m]{x^n} = x^{n\frac 1m}=x^{\frac mn}=x$ if $m=n$. Let's take $x=-2$, and $m=n=2$. Now we have, $\sqrt[2]{(-2)^2}=\sqrt[2]{4}=2$ But according to that ...
-1
votes
4answers
177 views

Is $ (((\sqrt{2})^ \sqrt{2})^ \sqrt{2})^{\cdots} $ an irrational number?

It is well known that $ \sqrt{2} $ is an irrational number. Is there someone who can show me if this number: $$ \left(\left(\left(\sqrt{2}\right)^ \sqrt{2}\right)^ \sqrt{2}\right)^{\cdots} $$ is ...
-5
votes
1answer
62 views

How to solve an exponent equation? [closed]

Can someone help me with this exponent equation. $2 ^ {2x+2} - 6 ^ x - 2 \times 3 ^{2x+2} = 0$ Any ideas how to solve it? Please I really have no idea. And I have exams. Tomorrow. And now it's like ...
-1
votes
1answer
59 views

What is the value of i^i? [duplicate]

i is an imaginary number. What is $i^i$? I tried to use euler rule but the answer is strange. For example $i = e^{\frac{1}{2}i\pi}$. Using $(a^b)^c = a^{b*c}$ we got $i^i=e^{(\frac{1}{2}i\pi)*i} ...
8
votes
5answers
415 views

How to solve this equation for $x$?$\left(\sqrt{2-\sqrt{3}}\right)^x + \left(\sqrt{2+\sqrt{3}}\right)^x = 2$

This is probably such a beginner question (and it's not homework). I've stumbled upon this: $$\left(\sqrt{2-\sqrt{3}}\right)^x + \left(\sqrt{2+\sqrt{3}}\right)^x = 2$$ How to solve this equation for ...
1
vote
1answer
37 views

What are the rules of powers of powers? [duplicate]

What would $2^{3^4}$ equate to? I can think of two rules that may apply: $a^{b^c} = a^{(b^c)}$ (Making $2^{3^4} = 2^{81}\approx2.417\cdot10^{24}$) or $a^{b^c} = (a^b)^c = a^{bc}$ (Making $2^{3^4} = ...
0
votes
1answer
128 views

What is the formula for summation of $n^n$? [closed]

How should I calculate: $$1^1+2^2+3^3+\dotsb +n^n$$ What is the formula for this submission?
2
votes
1answer
27 views

How to determine efficiently if the arithmetic addition and subtraction of certain powers of N can be equal to M?

I am given a number N and another number M . I have to find out if arithmetic addition and subtraction of certain distinct powers of N can lead to formation of number M . I tried different approaches ...
1
vote
5answers
77 views

Comparing two large numbers

Can you compare two large exponential numbers, like $5^{44}$ and $4^{53}$ without taking their logs?
1
vote
2answers
34 views

Combinatorial proof for summation of powers of two

I apologise if this has been posted before, but I've been poring over this problem for days now and just can't seem to get it. I'm looking for a combinatorial proof for: $2^n - 1 = 2^0 + 2^1 + 2^2 + ...
6
votes
2answers
172 views

How to find out the greater number from $15^{1/20}$ and $20^{1/15}$?

I have two numbers $15^{\frac{1}{20}}$ & $20^{\frac{1}{15}}$. How to find out the greater number out of above two? I am in 12th grade. Thanks for help!
1
vote
2answers
235 views

How is N^2/3 equivalent to 1/(N^1/3)?

I've tried to look for similar things on StackExchange and elsewhere on the net, but can't seem to find anything, so thought I'd just ask for some help on here... Someone has kindly helped me with a ...
1
vote
2answers
36 views

Converting a cryptographic hash to a string of English words: how many words are needed? (need help with exponentials)

A particular cryptographic hash is represented as a $57$ byte string, encoded as base $64$. RWSvUZXnw9gUb70PdeSNnpSmodCyIPJEGN1wWr+6Time1eP7KiWJ5eAM I want to ...
3
votes
0answers
34 views

Mapping exponential functions in polar coordinates

I tried mapping power functions onto the polar plane (i.e. converting x,y into r and $\theta$). I was successful with power functions representing $y=ax^n$ by $$r=\sqrt[n-1]{\frac ...
1
vote
1answer
132 views

How to minimize $a \times b$ where $a^b≥x$?

For example, if $x$ is 1 billion, the smallest possible $a \times b$ will be $3 \times 19 = 57$. This is because: $2^{30} \ge 1000000000$ $2 \times 30 = 60 $ $3^{19} \ge 1000000000$ $3 \times 19 ...