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

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5
votes
2answers
99 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
38 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
25 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= \sum_{\alpha<\beta}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
53 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
30 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
67 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
50 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
766 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 $x=...
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 = \dfrac{...
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
45 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
20 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
54 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
44 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
24 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
69 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, 16,...
3
votes
3answers
130 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? Here'...
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
36 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
66 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 $l^{-1}...
3
votes
2answers
72 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.
6
votes
2answers
147 views

Why doesn't $z^n\cdot\left(\frac{a+b}{z}\right)^n = (a+b)^n$ always hold?

When I entered is(z^n*((a+b)/z)^n = (a+b)^n); into Maple, the output was false and I guess Maple assumes that $a,b,n$ and $z$ ...
1
vote
4answers
47 views

Inverting Modular Exponentiation

How can I go about solving the equation $4 = y^4 \bmod{7}$? Do I have to try all of the possible $y$'s in between $1$ and $7-2$ or is there a smarter way that can be generalized for larger numbers?
1
vote
0answers
42 views

Convergence of power tower $(1/2)^{(1/3)^{\dots (1/n)}}$

I was wondering whether it is possible to evaluate the following limit $\lim_{n \to \infty} a_n = \lim_{n \to \infty} \frac{1}{2}^{\frac{1}{3}^{\kern3mu\raise1mu{.}\kern3mu\raise6mu{.}\kern3mu\...
1
vote
3answers
44 views

Determine whether a number is a power of 3

Why does $\frac{\log(n)}{\log(3)}$ being an integer determine whether $n$ is a power of three? While doing some programming exercises I came across this problem and the above formula was a proposed ...
1
vote
2answers
30 views

Determine all the possible values of $(\sqrt{3}+i)^i$ and specify which quadrant(s) of the plane contains these values.

$$ \mathrm{used}: a^z = \exp(z\log a)$$ $$(\sqrt{3}+i)^i = \exp(i\log((\sqrt{3}+i))$$ $$(\sqrt{3}+i)^i = \exp(i(\ln2+i(\frac\pi6+2k\pi)))$$ $$\mathrm{use}: e^z = e^x(\cos y+i\sin y)$$ $$ = (\exp(\...
2
votes
4answers
75 views

Prove that $\frac{p}{q}$ is a rational number with a finite decimal expression if $p$ is an integer and $q=(2^n)(5^m)$

Let $p,q$ be two integers and $q=(2^n)(5^m)$. Then $\frac pq$ is a rational number with a finite decimal expression. Any ideas how to do this? I've been thinking about it all day but I have no idea ...
3
votes
3answers
41 views

First digits of a cube of a natural number

Can a cube of a number be of form: $2016a_1a_2a_3\dots a_n$? I have no direction, and would love to get a certain direction/proof. Thanks in advance
1
vote
0answers
67 views

Axiomatic definition of exponent

Here's something I'm thinking about for a while, and would like to get feedback and some relevant references. Say I want to define by axioms an operation that will act like exponent, without using ...
3
votes
1answer
62 views

Finding the solutions of $n! \ge n^a$

Let $a \in \mathbb{N}, a \ge 2$ be a fixed natural number. Consider the inequality: $$n! \ge n^a$$ It can be proven that this inequality is true for sufficiently large values of $n$, but how can we ...
3
votes
1answer
109 views

Why is exponentiation right associative? [duplicate]

From Wikipedia: In order to reflect normal usage, addition, subtraction, multiplication, and division operators are usually left-associative while an exponentiation operator (if present) is ...
3
votes
2answers
96 views

Expressing Factorials with Binomial Coefficients

Expression I have somehow stumbled upon this expression (I believe I have proved it, but that is not important right now), which I have tried to simplify by writing it like something like this (I ...
2
votes
2answers
100 views

How is Faulhaber's formula derived?

I have been wanting to understand how to find the sum of this series. $$1^p + 2^p + 3^p +{\dots} + n^p$$ I am familiar with Gauss' diagonalised adding trick for the sum of the first $n$ natural ...
0
votes
1answer
33 views

Indices solving for $x$

Sorry this is a simple question but I'm having difficulty with it . $$2(16^{3x+2}) = 1 / 8^{5x-4}$$ I'm told to solve for $x$. My working - $$2(16^{3x+2}) = 1 / 8^{5x-4}$$ $$32^{3x+2} = 8^{-5x+4}...