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

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-4
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1answer
47 views

What is infinity to the zeroth power? [on hold]

I am not happy with the answers posted to similar questions. For example, in: What is infinity to the power zero the accepted answer is 1, which is definitely wrong. I think the answer is any ...
0
votes
0answers
11 views

Exponent operation over elements of $G$

I found a definition of an exponent operation over the element of $\mathbb{G}$ in this paper (page 4): $$ (g^a)^{\% b} = g ^{a \text{ mod } b}$$ I couldn't understand the rest of the paper (Decrypt ...
0
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0answers
18 views

Exponentiation of Gell-Mann Matrices

The Pauli Matrices satisfy the relation $e^{ia(n\cdot\sigma)}=I\cos(a)+ ia(n\cdot\sigma)\sin(a) $. Can a similar equation be derived for the Gell-Mann matrices? The main feature of the Pauli ...
1
vote
1answer
30 views

Subtraction of large numbers with exponents

Is there an easy way to break down the below formula? And make it easy to calculate it mentally without the use of a calculator? $108^2 - 92^2$ I know this is probably very basic, but I cant ...
5
votes
1answer
74 views

Combinatorics problem that deals with trigonometric functions

If $m$ and $p$ are positive integers and $m \geq p$, then show that $${m \choose 0}+{m \choose p}+{m \choose 2p}+{m \choose 3p}+\cdots$$ has value $${2^m \over p}\left(1+\sum_{k=1}^{\left ...
1
vote
1answer
17 views

negative fraction exponent and division

Quick question on how to handle negative fraction exponents when differentiating: I have this problem to differentiate. $$x^{2/3} + y^{2/3} = 1$$ So my textbook and I both did the first thing the ...
0
votes
1answer
44 views

How to calculate a definite integral with complex numbers involved?

I'm trying to calculate this integral, and I find it difficult when coping with complex numbers. $$ f(k) = \int_{lnK}^{\infty} e^{ikx} (e^{x}-K) dx ...
0
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1answer
32 views

Explanation of i to the i power? [duplicate]

Could somebody give me a good explanation for how $i^i$ works? I'm a junior and just now getting to this. I'm also too hard pressed for time to dive into exploring it myself.
1
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1answer
22 views

$2^i \equiv 2^j \pmod n$ implies $2^{j−i }\equiv1$ if $n$ is odd; also if $n$ is even?

Show that, if $0 \leq i < j$ and $2^i \equiv 2^j \pmod n$ and $n$ is odd then $2^{j−i} \equiv 1 \pmod n$. Is this necessarily true if $n$ is even? I have tried to prove this by using Fermat's ...
2
votes
1answer
15 views

How to find the highest [natural] radix base of a given number with a natural output

Like the title says, I'm trying to make a program that finds the highest natural radix of a given number with a natural output. My program works, but it loops every number possible number up to a ...
4
votes
0answers
175 views

What is the middle digit of $9^{99}$?

Find the total number of digits and the digit in the middle of $9^{99}$, $\it{without}$ actually calculating any other digit of the number. PS: according to defuse.ca/big-number-calculator.htm, the ...
2
votes
3answers
151 views

$(-1)^{\sqrt{2}} = ? $

This popped up when I was thinking about $$(-1)^{\frac {p}{q}}$$ where $ p $ and $q$ are integers such that $\gcd (p,q) = 1$ If $p$ is even : $(-1)^{\frac {p}{q}} = +1$ If $q$ is even : ...
0
votes
1answer
27 views

Powers with complex/negative bases

If x can be a positive real number (for example a fraction with a numerator and denominator), then why does the following relationship hold true only if and only if a and b are strictly positive real ...
1
vote
1answer
33 views

Limit with logarithm: $\lim_{n \to \infty} \frac{n^\alpha}{\ln^\beta n}$

What is the limit $\lim_{n \to \infty} \frac{n^\alpha}{\ln^\beta n }$ (ln=natural logarithm) for alfa real and less than zero? I found out it is zero for $\beta\ge0$, since then you can use the ...
1
vote
1answer
37 views

The exponent of $11$ in the prime factorization of $ 300!$ is___.

The exponent of $11$ in the prime factorization of $ 300!$ is $27$ $28$ $29$ $30$ My attempt: According to Exponent of $p$ in the prime factorization of $n!$ ...
1
vote
1answer
22 views

Digit-sum division check in base-$n$

Several years ago now I realised that for any natural numbers $x$ and $y$ you could write $$x^y=(x-1) \left(\sum_{i=0}^{y-1}x^i\right)+1$$ This shows that $x^y-1$ will always be divisible by $x-1$, ...
0
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0answers
18 views

Exponent of x in the prime factorization of y?

This is a simple one. I recently came across the following phrase: "$(x)_y =$ the exponent of $p_y$ in the prime factorization of $x$, for some $x, y<0$" I know what prime factorization is, and ...
-5
votes
1answer
57 views

Proof that $0^0 \neq 1$ [closed]

Suppose that $t = \sqrt{t}^{\sqrt{t}}$, then, it follows that; $$ t^{\sqrt{t}} = \sqrt{t}^{t} \\ \frac{1}{2}t\ln{\left(t\right)} = \sqrt{t}\ln{\left(t\right)} \\ \ln{\left(t\right)}\left[\frac{1}{2}t ...
0
votes
1answer
9 views

Name for inequality about sums of exponents with same base

Is there a name for the following inequality regarding sums of exponents which share a base? $$\text{For all integers $b \geq 2$, $n \geq 1$,} \\ \sum_{i=0}^{n-1}{b^i} < b^n$$
0
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0answers
13 views

Is it possible to transform different interest rates (applied over different times) into one interest rate?

I have a fixed income asset that has the following payment schedule. Is it possible to transform the multiple interest rates being applied to different periods into one interest rate and one period? I ...
3
votes
2answers
56 views

Enough differences of powers of natural numbers equals a constant

Let $n_k$ be a sequence. We create a new sequence by taking the difference of consecutive terms in $n_k$. So the terms of the new sequence $a_k$ is defined as $a_i=n_{i+1}-n_i$. This is the difference ...
2
votes
1answer
49 views

Scalar by matrix derivative $\frac{d {\rm tr} (e^{\bf X} {\bf A})}{d {\bf X}}$

I'm trying to figure out what is the scalar by matrix derivative $\frac{d {\rm tr} (e^{\bf X} {\bf A})}{d {\bf X}}$ equals to? I know that $\frac{d {\rm tr} (e^{\bf{X}})}{d \bf{X}} = e^\bf{X}$ and ...
0
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3answers
31 views

Exponentiation of Diagonalizable Matrix

Wikipedia says that "If $A = UDU^{−1}$ and D is diagonal, then $e^{A} = Ue^{D}U^{−1}$" Why is this the case? I understand that $e^D$ yields a matrix where $M_{i,j} = e^{D_{i,j}}$, but how is it ...
1
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4answers
47 views

Simplifying and taking the limit

I would like to compute the following limit $$\lim_{n\to\infty}\frac{10^{n^{2}}}{10^{(n+1)^{2}}}$$ but I'm having a hard time simplifying. Can anyone explain to me the properties of these exponents? ...
0
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0answers
21 views

Given complex numbers raised to real powers, what do imaginary part = 0 level sets look like?

Suppose you lay flat (horizontally) the Argand plane, with x real. Consider the vectors v in that plane. Then let z, the vertical axis, be the real part of the complex solution to v^t, with t real. So ...
0
votes
1answer
37 views

How to solve an equation with the unknown variable both in the exponent and in linear summand?

How to solve the following equation? $$e^{dz} - mdz = 1$$ where $z$ is the unknown variable, the others are constants, $\exp(x)$ is taking $e$ to the power $x$ . I am interested in real solutions ...
0
votes
1answer
26 views

Simplifying Large Bases with large Exponents

I'm told to find: $105 308^{7125} \pmod {11}$ I'm not exactly sure how to go about calculating this. I know that I could split the exponent into multiples of it, for instance. $7125 = 7 * 10 * 10 * ...
0
votes
3answers
45 views

How to differentiate x^(1/x)?

How to differentiate the following? $$x^{\frac{1}{x}}$$ (I know the answer is $\frac{1-\ln(x)}{x^{2-\frac{1}{x}}}$, but I do not understand how to get there) Attempt at solution I believe the ...
0
votes
2answers
31 views

Order of exponents matters?

I always thought that $(a^b)^c=a^{bc}$... I am confused about why the order of exponents seems to matter in this particular case: $((x-0.5)^2)^{1/3}$ is not the same as $(x-0.5)^{2/3}$ ...
-1
votes
1answer
28 views

approximation of binomial coefficient by exponentiation

Show that inequality holds for every n, k. $$\ { n \choose k} \leq \frac{n^k}{k!}. ( \, 1-\frac {k}{2n} ) \,^{k-1} $$ I used Stirling's formula but I stucked, which is $$\ { n \choose k} = ...
5
votes
2answers
95 views

What is $2^{(k+1)} +2^{(k+1)}$ equal to?

I am so confused with this question. When the powers add we get $2^{2k+2}$ but in my book it says $2^{k+2}$. How is that? Please explain.
-1
votes
1answer
33 views

The 'unique' way powers are calculated

everyone. My question for you is why a power, such as m^n^o is calculated as m^(n^o), and not (m^n)^o. Let me explain why I don't fully understand this: As we know, we are supposed to perform ...
0
votes
1answer
65 views

What does $(-2)^x$ really mean?

I think we can all agree that $(-2)^{-1}=-1/2,(-2)^0=1,(-2)^1=-2,(-2)^2=4$ But what does the function $f=(-2)^x$ really mean? It is defined on the integers based on how most people understand ...
1
vote
1answer
22 views

Cardinal exponentiation question of infinite cardinals.

I got a little confused with a question about cardinal exponentiation: Let $\beta$ be an ordinal, and let $K_\alpha$ , $\alpha < \beta$, be infinite cardinals with $K= ...
4
votes
2answers
82 views

Find $x$ in the equation $x^x = n$ for a given $n$

Simply: How do I solve this equation for a given $n \in \mathbb Z$? $x^x = n$ I mean, of course $2^2=4$ and $3^3=27$ and so on. But I don't understand how to calculate the reverse of this, to get ...
0
votes
1answer
45 views

How many positive integers x and y satisfy the equation $x^y = (2016)^{2016}$?

How many positive integers x and y satisfy the equation $x^y = (2016)^{2016}$ ? Explain your answer. I started by factoring $2016$. I found the factors to be $36$, but I couldn't go further.
0
votes
2answers
28 views

Fractional Exponents - Is the sign discarded?

For example, 16^(3/4) Is the accepted as both -8 and 8 or just 8? I ask this because on an AS maths mark scheme it says to condone -8 Thanks
0
votes
1answer
41 views

What does $x^{(i)}$ Mean or Denote

I know this is a simple question, but what does $x^{(i)}$ mean (where $x$ and $i$ are variables and $i$ isn't $\sqrt{-1}$) or what operation does it denote? I assume it's not a regular exponent. I saw ...
0
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0answers
27 views

How can you distinguish modular exponentiation from random?

Let $N$ be the product of two primes and let $P$ be the smallest prime larger than $N$. Let the algorithm $R(N,s)$ return $s^{1/P} \pmod{N}$. Let the algorithm $\widehat{R}(N,s)$ pick a ...
2
votes
3answers
64 views

On the integral of $e^{aix}$.

I've been trying a simple trick, but I'm unsure as to why it is failing. Let's say I wanted to compute $ \int^{\pi} _{ - \pi} e^{aix} dx$ for some constant a. Why won't this trick work? $ \int^{\pi} ...
1
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3answers
25 views

Expression evaluation

I have encountered this expression and I cannot evaluate to the desired result on the right side $$3\cdot4^{k+1}+4^{k+1}-64 = 4^{k+2}-64$$
1
vote
3answers
47 views

Solving $X x^t+ Y y^t=1$ for a specific case with constraints

Is there an analytical solution to the equation below? $$\begin{align*} A\frac{\alpha}{k}e^{(k-\alpha)t}+B\frac{\beta}{k}e^{(k-\beta)t}=1 \end{align*}$$ where $\alpha$ and $\beta $ are roots of ...
2
votes
2answers
62 views

Proof that $2^{2^x}$ ends in 6

So I just checked every number of the form $2^{2^x}$ up to $2^{8192}$ and they all end in $6$. Can someone formally prove that this will be true for all $x$?
1
vote
1answer
14 views

formula for equation with exponent variable?

is there a closed formula for such an equation to find the value of $x$ in $ax = b^x$ if there isn't , are there any published attempts ?
5
votes
4answers
754 views

Why is it more efficient to compute the modular exponentiation by calculating to the power of two and not three for example?

I learned about modular exponentiation from this website and at fast modular exponentiation they calculate the modulo of the number to the power of two and then they repeat this step. Why not ...
1
vote
2answers
101 views

Confusion regarding taking the square root given an absolute value condition.

From the Generating function for Legendre Polynomials: $$\Phi(x,h)=(1-2xh+h^2)^{-1/2}\quad\text{for}\quad \mid{h}\,\mid\,\lt 1$$ My text states that: For ...
2
votes
1answer
29 views

Solving for the exponent of a power sum

Let $x$, $y$, $z$, $t$, be real positive numbers. Is it possible to solve $t$ from the following equation, if $x$, $y$, and $z$ are known? $$ x^t + y^t=z $$ If an exact solution is not possible, are ...
0
votes
1answer
24 views

What happens when I increase the exponent and decrease the base?

Let $y_1=r^n$, where $r>1$ and $n>1$. Suppose we decrease $r$ and increase $n$ such that $y_2=(r-\epsilon)^{n+\beta}$. If $\epsilon>\beta$, can we prove that $y_2<y_1$?
1
vote
1answer
42 views

Prove that the two powers are equal

Prove that: $$\dfrac{1}{2^{180}a^{360}}\dfrac{(a^{720}-1)(a^2-1)}{a^{2}+1} = \dfrac{\left(1+\dfrac{\sqrt{3}}{2}\right)^{180} - \left(1-\dfrac{\sqrt{3}}{2}\right)^{180}}{\sqrt{3}}$$ where: $$a = ...
0
votes
1answer
34 views

Logarithmic to linear

Given this function: $$\frac{1.0}{1024.0} + \frac{x}{100.0} * \frac{1023.0}{1024.0} = y$$ $$10 * \frac{\log_{10}(y)}{\log_{10}(2)} = z$$ $$z * 100 = a$$ ...