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

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-5
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1answer
59 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
10 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
votes
0answers
14 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
61 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
votes
3answers
36 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 ...
2
votes
4answers
52 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
votes
0answers
23 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
32 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 $(x-0.5)^{2/3}$ ...
0
votes
1answer
31 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
97 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
35 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
66 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
23 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
83 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
49 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
votes
0answers
28 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
66 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
vote
3answers
26 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
48 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
758 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
30 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
25 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: $$\frac{1.0}{1024.0} + \frac{x}{100.0} * \frac{1023.0}{1024.0} = y$$ $$10 * \frac{\log_{10}(y)}{\log_{10}(2)} = z$$ $$z * 100 = a$$ ...
1
vote
3answers
44 views

Show that there are infinitely many positive integers A such that 2A is a square, 3A is a cube and 5A is a fifth power.

Show that there are infinitely many positive integers A such that 2A is a square, 3A is a cube and 5A is a fifth power. Using some arithmetic, I felt that if $A = 2^{15k}3^{20k}5^{24k}$ then it ...
1
vote
1answer
19 views

Term for using a number as an exponent (complementing “raise to the power of …”)

If we have an equation we want to solve such as $\sqrt{x} = 3$, we can say something such as square both sides or "raise both sides to the power of 2", to arrive at $x = 9$. So $3 \rightarrow 3^2$ ...
2
votes
2answers
46 views

Compute the indicated power of a matrix

Compute the indicated power of the matrix: $A^8$ $ A = \begin{bmatrix}2&1&2\\2&1&2\\2&1&2\end{bmatrix} $ I calculated the eigenvalues: $ \lambda_1 = \lambda_2 = 0, \lambda_3 ...
0
votes
1answer
32 views

Summation of terms of an exponential progression.

I was recently considering a progression where each term in the sequence is the previous term raised to a common exponent. To elucidate: $$S_{E.P}(a,m)=a,a^m,{(a^m)}^m,({(a^m)}^m)^m \cdot \cdot ...
0
votes
4answers
43 views

How to compute $3^9 \pmod {10}$

The solution given was: $3^9 = (27)^3 = 7^3 = 49(7) = 9(7) = 63 = 3 $ I understand up to $\ 3^9 = (27)^3 $ But after that I am lost. Can someone explain how to solve this and what is going on ...
-1
votes
1answer
23 views

Exponentiation Function - Real root

I have a function with below format: $f(x)$ = $A x ^ {-.1} + B x ^{ -.5} - C$ where: $ A,B,C> 0 $ How can I check if the function has real roots? the domain of $x$ is: $ x\in\mathbb{N} $ ...
2
votes
1answer
34 views

How can I express such a product?

I know for example that $$\prod^{k}_{n=0} a_n = a_0 \cdot a_1 \cdot a_2 \cdot a_3 \cdots a_k$$ But what if I wanted to express $\space 3^k$ as a product? I know it sounds like a simple question, ...
2
votes
3answers
51 views

Can someone explain why this happens? (Dividing variables with exponents)

Alright, so let's say I have $$\frac{x^{-6}}{-x^{-4}}$$ The answer is $\dfrac{1}{x^2}$, but why isn't it $\dfrac{1}{-x^2}$?
0
votes
6answers
68 views

How many digits are in the integer representation of 2 to the 30th power?

How many digits are in the integer representation of 2 to the 30th power? Since I didn't really know any 'expert' way to approach this, I just started out by listing the powers of 2, like 2, 4, 8, ...
3
votes
3answers
129 views

Wolfram Alpha wrong answers on $(-8)^{1/3}$ and more? [duplicate]

Wolfram Alpha doesn't give $-2$ for $(-8)^{1/3}$, and it absolutely fails to draw $f(x)=x^{1/3}$ - does anyone know why? Am I missing something very 'deep' Wolfram Alpha is trying to teach me? ...
0
votes
1answer
20 views

What exactly are narrower and wider conditions? (Polya's How To Solve It)

In How To Solve It, on page 56, Polya states that "If we pass from a proposed condition to a new condition equivalent to it, we have the same solutions. But if we pass from a proposed condition to ...
-2
votes
1answer
33 views

Calculating non-integer exponents

I'm trying to create a fully functioning calculator in Ruby and have been stuck for quite some time on calculating non-integer exponents, e.g. $3^{2.4}$. I would like to be able to support decimal ...
3
votes
2answers
61 views

How to solve simultaneous exponential equations with polynomial parts?

I have been puzzled at how to simultaneously solve the following equations and before I give up entirely I thought I'd turn to fellow mathematicians first: $$2^x=3y$$ $$2^y=5x$$ I have graphed both ...
0
votes
1answer
25 views

find the proper matrix of exponetial part

I am awkward to calculate the matrix so I would like to get some help $exp(y^{T}V^{T}\Sigma^{-1}S_{X}- \frac{1}{2}y^{T}ly)$ is proportional to $exp(- \frac{1}{2}(y-a)^{T}l(y-a))$ and $a$ is ...
3
votes
2answers
68 views

How fast does this sequence grow?

I have the following recursive definition of a sequence of numbers: $$a_{n+1}=(a_n)^{(a_{n-1})}$$ And $a_0=a_1=2$. The first few terms are: $$a_2=4$$ $$a_3=16$$ $$a_4=65536$$ $$a_5=1.1579209 ...
9
votes
7answers
2k views

Is there a way to calculate absurdly high powers? [closed]

Could it be at all possible to calculate, say, $2^{250000}$, which would obviously have to be written in standard notation? It seems impossible without running a program on a supercomputer to work it ...
0
votes
1answer
46 views

Solve $x^y+y^x=a$ for $y$?

Just as I question states, I want to solve the equation for $y$, but that is proving difficult as you cannot simply just use algebraic methods. I suspect the Lambert W function might come into play.