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

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2
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2answers
63 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
775 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
31 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
26 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
43 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
46 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
21 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
59 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
37 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
71 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
133 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
22 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
39 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
70 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
27 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
77 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
43 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
31 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
76 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
44 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
68 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
137 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
99 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
3answers
137 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}...
1
vote
1answer
18 views

Simplifying $(-2x^2 y^{-1} )^3 / 2y^{-2}$, with positive exponents

The question is $$(-2x^2 y^{-1} )^3 / 2y^{-2} $$ I did it till $$-4x^6 y^{-1}$$ I'm told to put it in a form of positive indices . Can I say that the answer is $x^6 / 4y$ ?
0
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0answers
32 views

Good reference on the parametrization of SU(3) and SU(N)

For the 2-dimensional $SU(2)$ matrices, there is a fairly general parametrization formulation: $s_2=\begin{bmatrix} e^{i\alpha}cos(\theta) & -e^{-i\beta}sin(\theta) \\ e^{i\beta}sin(\...
0
votes
1answer
32 views

Is this equation on exponentiation true?

Is the below equation true for all real values of $a$, $b$, $c$ and $d$? (with $c\neq0$) $$d(a+b)^c= (ad^{1/c}+bd^{1/c})^c$$ Some cases has been tested and the above has not been disproved.
0
votes
0answers
27 views

Derivative of expresion with matrix in exponent

I am looking at a proof which states the following results: $$\frac d {dt}\left(A \frac{t^2}{2!} \right)=A^2t$$ where $A$ is a matrix. Why is it that the matrix gets squared, and for the derivative ...
7
votes
4answers
125 views

How to define $x^a$ for arbitrary real numbers $x$ and $a$

Questions like this, which asks to solve $$x^{\frac43} = \frac{16}{81}$$ confuse me. The solution for real $x$ is $\pm \frac8{27}$. The question would presumably be a bit different to solve $$x^{\...
2
votes
1answer
56 views

Proof of part of properties of exponentiation Tao proposition $4.3.12$

If you let $x,y$ be non-zero rational numbers, and let $n, m$ be integers, I need to prove that if $x \geq y>0$, then $x^{n} \geq y^{n}>0$ if n is positive, and $0< x^{n} \leq y^{n}$ if $n$ ...
2
votes
1answer
69 views

Is a real logarithm of a special orthogonal matrix necessarily skew-symmetric?

The exponential map from the Lie algebra of skew-symmetric matrices $\mathfrak{so}(n)$ to the Lie group $\operatorname{SO}(n)$ is surjective and so I know that given any special orthogonal matrix ...
0
votes
3answers
128 views

Unspecified $x^y$ vs. $y^x$ - which is larger?

Given only the expressions $x^y$ and $y^x$ and no additional information except $x\neq y$ (and the meta-knowledge that the problem was presented in the context of induction), is it possible to ...
1
vote
0answers
22 views

Alternative to exponents

In a loose sense addition is repeated successorship, multiplication is repeated addition, and exponentiation is repeated multiplication. However, the latter is the only one that isn't well defined ...
2
votes
1answer
60 views

Proving $a^b$ is well defined

How do I prove that $$\lim_{(m,n) \to \infty} a_m^{b_n} = a^b$$ where $a,b \in \mathbb R$, $a_i,b_i \in \mathbb Q$, $a_m \to a$, $b_n \to b$ and $a$ and $b$ are not both zero, and $a_m >0$ I can ...
2
votes
3answers
58 views

Does the complex modulus satisfy the power identity $|z^r|= |z|^r$?

Can we "split the modulus" of complex numbers? Let $z\in\mathbb{C}$. Then, does $$|z^{r}|=|z|^r$$ hold, where $r\in\mathbb{R}$. Is this true even for $r\in\mathbb C$ ? Also, can we show this? I am ...
-2
votes
1answer
52 views

Proof: Raising a complex number to a rational power

The problem from the textbook is: Prove that if (a complex number) $z$ is a number on the unit circle, then $z$ has finitely many distinct powers $z^n$ if and only if the argument of $z$ is a ...