Questions about exponentiation

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2
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0answers
79 views

Fast check If the remainder is 1

Is there any fast method to 'say' that $R = (A \mod B)$ is $1$ or $R > 1$ or $R \neq1$ or $R > k>1$ ( where $k$ is a small integer on $32$ bits) without to actually calculate the real value ...
0
votes
1answer
34 views

Prove that $\forall \, a,b \in \mathbb{N}- \{0,1\}\,\, \wedge \,\,a<b \,\, ; \,\, a^{1/a} > b^{1/b}$

Prove that $\forall \, a,b \in \mathbb{N}- \{0,1\}\,\, \wedge \,\,a<b \,\, ; $ $$\,\, a^{1/a} > b^{1/b}$$ I need some tip to start it. Thank you.
0
votes
2answers
221 views

Evaluate a matrix with a negative power

I am having problem with how to calculate a matrices that are raised to negative powers. I can manage the adding, multiplication etc, but I am stuck here. The matrix in question is ...
5
votes
2answers
119 views

How many values does $1^{\alpha}$ have for $\alpha$ irrational?

One such value is $\displaystyle\cos\left(2\pi\alpha\right)+i\sin\left(2\pi\alpha\right)$, by Euler's theorem. On the other hand, we can choose an arbitrary sequence $S=(a_n)_n$ of rational numbers ...
2
votes
2answers
80 views

Differentiation of exponential function? [closed]

How to solve derivative $\lim_{n\to\infty}e^{{}^n(x)}$ with respective of $x$ ? Here, ${}^n(x)$ is a tetration function $$ {}^n(x)= \begin{cases} x^{[{}^{n-1}(x)]} & \mbox{ if } {\;n>1}\\ x ...
0
votes
3answers
110 views

Find $x ^{ 2013} + 2013x ^{ 2010}$

Q. If $\large {\space x^2 + x + 1 = 0\space } $, Find $ x ^{ 2013} + 2013x ^{ 2010}$. I have tried finding the roots of $x$ from the given equation but that does not work.
3
votes
4answers
131 views

What is the shortest way to compute the last 3 digits of $17^{256}$?

What is the shortest way to compute the last 3 digits of $17^{256}$ ? My solution: \begin{align} 17^{256} &=289^{128} \\ &=(290 - 1)^{128}\\ &=\binom{128}{0}290^{128} - ... ...
1
vote
1answer
28 views

Negative powers in modular arithmetic

Suppose we have set $Z = \{0, 1, \dots, N-1\}$ with arithmetic operations modulo $N$; $a > 0$ is an element in $Z$. Is it possible that $a^{-1}$ does not exist but $a^{-n}$ exists for some $n$, $1 ...
6
votes
2answers
59 views

What are the number of integers a $1 \le a \le 100$ such that $a^a$ is a perfect square.

What are the number of integers a such that $1 \le a \le 100$ and $a^a$ is a perfect square. I think the answer should be 51 since a can be 1 and then 2,4,6,...100. Is the answer correct?
8
votes
5answers
120 views

Solving equations of type $x^{1/n}=\log_{n} x$

First, I'm a new person on this site, so please correct me if I'm asking the question in a wrong way. I thought I'm not a big fan of maths, but recently I've stumbled upon one interesting fact, which ...
0
votes
0answers
65 views

difficult inequality to prove

I need help proving this inequality is correct for a homework assignment: $$\displaystyle \left(\frac{13}{4}\right)^{n} \leq ...
2
votes
3answers
99 views

How can we differentiate $(x^{-1})^{({x^{-1})^{x^{-1}}}}$ wrt $x$?

How can we differentiate $(x^{-1})^{({x^{-1})^{x^{-1}}}}$ with respect to $x$?
6
votes
1answer
200 views

What does the raised $^2$ stand for?

What does the raised $2$ stand for? My first guess was: $4^2$ is $2\times 4=8$? Note: Am not really good at math
5
votes
2answers
135 views

Non-integral powers of a matrix

Question Given a square complex matrix $A$, what ways are there to define and compute $A^p$ for non-integral scalar exponents $p\in\mathbb R$, and for what matrices do they work? My thoughts ...
0
votes
4answers
49 views

Is the power of 1/2 same thing as principal square root?

$\sqrt{9} = 3$ 9 has 2 square roots: 3 and -3. What is $9^\frac12$? Is $9^\frac12 = \sqrt{9} = 3$ or is $9^\frac12 = \pm3$?
1
vote
1answer
45 views

Is there a formula for a sequence like $k^{t}-k^{t-1}+k^{t-2}-…+k^{2}-k^{1}+k^{0}$

I am trying to solve a programming problem and my intended solution involves a calculation like this one: $k^{t}-k^{t-1}+k^{t-2}-...+k^{2}-k^{1}+k^{0}$ The problem is that $t$ can be as large as ...
1
vote
1answer
42 views

Solving for exponent with multiple bases

From a practical perspective, my question can be most easily considered as solving for time in a future-value type equation, but for two separate investments growing at different rates. Say you have ...
2
votes
0answers
71 views

Prove $a^b=\underbrace{a \times a \times \cdots \times a}_\mathrm{b\ times}$

Starting with the definition $a^b = \exp(b \ln a)$, where $\exp$ is defined by its Maclaurin series and $\ln$ is defined by the integral $$\ln x = \int_1^x \frac{dt}{t}$$ how can we show that ...
0
votes
1answer
24 views

Pandigital exponent solution?

I don't have Reviewing C++ By Alex Maurea book. But someone in facebook post a question from this book which is following You can see the puzzle at page 542 problem 29. Now I think the answer ...
1
vote
4answers
145 views

Which is larger :: $y!$ or $x^y$, for numbers $x,y$.

This is a generalization of this question :: Which is larger? $20!$ or $2^{40}$?. No explicit general solution was presented there and I'm just curious :D Thank-you. Edit :: I want a most-general ...
2
votes
1answer
34 views

Problem finding limit - which function is asymptotically larger

I have a homework question, so please don't answer fully but I would appreciate a push in the right direction. Basically we need to figure out if $n^{n+\frac{1}{2}}e^{-n}$ is larger,smaller, or equal ...
3
votes
4answers
98 views

Calculating $\log_7 125$

So the problem asks to calculate $\log_7 125$. It's multiple choice and the options are $2.48$ $4.75$ $1.77$ $2.09$ Given that $7^2 = 49$ and $7^3 = 343$, the answer must be either option 1 or 4, ...
6
votes
6answers
210 views

What is $(-1)^{\frac{2}{3}}$?

Following from this question, I came up with another interesting question: What is $(-1)^{\frac{2}{3}}$? Wolfram alpha says it equals to some weird complex number (-0.5 +0.866... i), but when I try ...
0
votes
1answer
28 views

Fractional exponentiation in modular arithmetic

Does raising a modular expression to a fraction mean anything? For example, $a\,\,mod \,\,N$ raised to $1/b$ where $b>0$. Does this violate the rules of modularity?
6
votes
2answers
94 views

What's the first digit of 2410^2410?

The first digit means the left most digit. 2410 is just an example and it can be replaced by any other numbers. Can any one help me to solve it?
-1
votes
1answer
48 views

How do I solve for $t$ and $s$ in $y = x^{-t/s}$?

I have $$y = x^{-t/s}$$ How do I solve for $t$ and $s$ in terms of the other variables?
1
vote
2answers
80 views

Why is $2^a > a^3$?

I found this rather interesting and maybe, a bit too obvious for some people property about 2 raised to some power. $2^a > a^3$, if $a=0,a=1 \text{ or } a\ge 10$ .($a \in N$) I seem to get a bit of ...
4
votes
4answers
85 views

negative exponent problem

$$\sqrt{\frac{1}{3^0 + 3^{-1} + 3^{-2} + 3^{-3} + 3^{-4}}}$$ Does this equal = $$ \begin{align*} & \sqrt{3^0 + 3^1 + 3^2 + 3^3 + 3^4} \\ =&\sqrt{1 + 3 + 9 + 27 + 81} \\ =&\sqrt{121} \\ ...
1
vote
4answers
69 views

What exponent should I raise $26$ to in order to equal $2^{76}$?

I want to figure out how long an all-caps password needs to be to equal $2^{76}$ bits of security. I would type this into Wolfram Alpha, but I'm not sure what function to use or if it can compute ...
1
vote
3answers
52 views

Exponential Equations

I solved this , but I am not sure if I did in the right way. $$2^{2x + 1} - 2^{x + 2} + 8 = 0$$ $$2^{x + 2} - 2^{2x + 2} = 8$$ $$\log_22^{x + 2} - \log_22^{2x + 2} = \log_28$$ $$x + 2- 2x - 2 = ...
0
votes
0answers
27 views

$A_{2}^{T} + A_{2} < 0$ for $A_{2} =(A_{1}A_{0}^{-1})^{\alpha}A_{0}$?

Given are two matrices $A_0, A_1$, whose symmetric part is negative definite: $A_{0}^{T} + A_{0} < 0$, $A_{1}^{T} + A_{1} < 0$ Proof that: $A_{2}^{T} + A_{2} < 0$ for $A_{2} = ...
1
vote
1answer
24 views

What contributes more to a increase in value of a exponentiation: a increase of base or exponent?

Firstly, my apologies for being a programmer messing in math land. I was wondering whether the value of a exponentiation is increased more by a increase in the base or exponent (I believe this is a ...
1
vote
2answers
46 views

Polar form for complex number with variable exponent

This might be an easy question but I'm having trouble showing that $\left(1+\frac{i\theta}{m}\right)^m$ has the angle $ m\arctan\left(\frac{\theta}{m}\right)$ in polar form on the complex plane. ...
12
votes
7answers
314 views

integral of $x^2e^{-x^2}~dx$ from $-\infty$ to $+\infty$

I know that the $$\int^{+\infty}_{-\infty}e^{-x^2}~dx$$ is equal to $\sqrt\pi$ It's also very clear that $$\int^{+\infty}_{-\infty}xe^{-x^2}~dx$$ is equal to 0; However, I cannot manage to ...
7
votes
2answers
191 views

Can $2^k + 2^j$ be expressed as $2^n$?

If we are given $j,k \geq 0, j> k$ and $j,k$ are integers, can $2^j + 2^k$ be ever expressed as $2^n$ where $n \geq 0$ and is an integer? What I said: Suppose it can. Then for some $0 \leq n ...
1
vote
3answers
42 views

Calculating $\phi(x^y)$

I know how to compute $\phi(x)$ like $\phi(21)$ or $\phi(7)$ but how can I compute $\phi(x^y)$. Specifically how can I compute $\phi(5^{20})$?
3
votes
3answers
91 views

Exponentials in complex numbers

If $\displaystyle z-\frac1z=i$, then find $\displaystyle z^{2014}+\frac{1}{z^{2014}}$. The answer should be in terms of $1, -1,\;i\;or\;-i$. I am not able to understand how to simplify the given ...
0
votes
1answer
49 views

Solve some unusual log/exponential equations

I understand about log and exponential equations/functions, but I can't solve these (the numbers are just examples, of course): $ 4^x = x + 10$ $x^x = 3$ $(2x + 3x^2)^{x + 1} = (x - x^3)^{x^2}$ Are ...
8
votes
6answers
314 views

Find $x$ and $y$ in $2^{x-y} + 1 = 2^x,$ where $x,y$ are integers

I have no idea what to do now. Is there any way to find the integers $x$ and $y$ by factoring? Thank you.
0
votes
1answer
73 views

Negative Base to non-integer power

I'm looking to consistently solve the m^n case, including conditions where m is negative and n is non-integer. I'd like to, additionally, catch the error when it isn't possible. Some examples to ...
5
votes
2answers
97 views

Extending exponentiation to reals

I've been reading through a course on exponential functions, starting from integer-valued exponents to rational ones as in: $x^r$ from $r\in \Bbb{N}$ to $\Bbb{Z}$, and combining them to rigorously ...
1
vote
2answers
59 views

Recursive definition of recursively defined operations

The recursive definitions of addition, multiplication, and exponentiation usually stop after exponentiation ("${\small+}1$" to be read as "the successor of"): $x \boldsymbol{+} (y\ {\small+}1) := (x ...
0
votes
4answers
93 views

Calculating St. Ives paradox. Finding the right solution.

There’s an old nursery rhyme that goes like this: As I was going to St. Ives, I met a man with seven wives, Each wife had seven sacks, Each sack had seven cats, Each cat had seven kits: Kits, ...
1
vote
2answers
110 views

How can you prove that a value raised to a fraction($\frac{1}{2}$ for example), is $\sqrt{x}$?

I know that if you raise a value to $\frac{1}{2}$ for example, you take the square root, but that is not what I am asking, what I am asking is; what are you actually doing when raising a value to ...
0
votes
1answer
20 views

How many people are included in three-hop network analysis? [closed]

According to the ACLU, if you have 40 contacts in your network, and the NSA collects data from your network to three degrees of separation, then it could collect data on up to 2.5 million people. How ...
0
votes
0answers
21 views

I need to find $n$ that $\frac{1}{(n+1) \cdot \ln(n+1)} <10^{-4}$

$\frac{1}{(n+1) \cdot \ln(n+1)} <10^{-4}$ So what I did is this: $(n+1)\ln(n+1) > 1000 \Rightarrow n>190$ When I put it back I see that $\frac{1}{192 \cdot ln(192)} \not < 10^{-4}$. ...
2
votes
3answers
61 views

Convert from high exponent of base $10$ to base $2$.

Is there an efficient way to convert from a high exponent of base $10$, to base $2$? Both in exponent notation. Here's an example: If I have a number that's $10^5$ or even $10^{100}$, and I wanted to ...
1
vote
1answer
48 views

Simplify an expression.

Don't know how to do this. Simplify the expression, show steps: $$\large \dfrac {a^{-\frac 14}a^{\frac 32}}{a^{\frac 13}}$$ Write the answer using only positive exponents. Assume that all variables ...
0
votes
1answer
16 views

A complex exponential equation

Find x in the following equation: (1+root(3))^x + 2^(x-1)*(2+root(3))^x = 4 I have no idea what this site's "quality" standards are, which are preventing me from posting the question, so I'm typing ...
3
votes
2answers
56 views

Is there a proof that $n^xm^x = (n^x)^{(\log(mn)/\log(n))}$?

This isn't a homework question, just something I'm curious about, but you can treat it that way if you like. So the other day I was playing with my calculator and I noticed that $$ 2^x10^x = ...