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Questions tagged [exponentiation]

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

3
votes
4answers
647 views

What is the best algorithm for finding the last digit of an enormous exponent? [duplicate]

I found most answers here not clear enough for my case such as $$ 123155131514315^{4515131323164343214547} $$ I wrote the $n\bmod10$ in Python and execution time ran out. So I need a faster ...
-2
votes
1answer
22 views

GCD and exponentiation of large numbers

I am solving a problem involving $\gcd$ of two very large numbers. Given three numbers $a,b,n$, I have to find $\gcd(a,b^n)$. So for example $$a,b,n=119929244861828206, 521483382396998375, ...
0
votes
2answers
36 views

Integral of Gaussian multiplied with a real valued power law

I want to calculate $$\int_{0}^{\infty}x^{\beta}\exp^{\frac{-(x-x_0)^2}{2\sigma^2}}dx $$ for continuous values of $\beta$ that are positive real (in my case, $\beta \in [0,10]$), and for $x_0 \in R^+$ ...
6
votes
3answers
391 views

Why does the approximation for exponents $(a+b)^c \approx a^{c-bc} (a+1)^{cb}$ work?

I was working with some code involving exponents in an environment where exponents can only be calculated if the base of the exponent is an integer. I needed a good fast way to approximate this ...
0
votes
1answer
73 views

Is $f(z)=2^z$ a homomorphism?

I'm new to complex variable analysis. In the real case, $f:(\mathbb{R},+)\to(\mathbb{R^+},\cdot)$ defined by $f(x)=2^x$ is a homomorphism, furthermore, a isomorphism. In the complex case, I wonder if ...
2
votes
1answer
148 views

Solve $i^{i^{i^\ldots}}$ [duplicate]

How to find $$i^{i^{i^\ldots}} \quad :\quad i=\sqrt{-1}$$ I'm able to find the solution for the finite powers using $$i=e^{i(2k\pi+\frac{\pi}{2})}\quad:\quad k\in\mathbb{Z}$$ $$i^{i}=e^{-(2k\pi+\...
1
vote
2answers
32 views

Solving for a variable in a complex algebra equation with logs and powers

I have a follow-up question to my last one. I need to solve for b for the following: $$d=\frac{s(\ln(o))^b}{s(\ln(g))^b}$$ Once again, this is beyond the level of ...
0
votes
1answer
29 views

Solving for a variable in an algebra equation with logs and powers

I'm working on a software project and I need to solve for b in the following equation: $$e^{\ln(gx)ab}-e^{\ln(x)ab}=c$$ I've tried a couple of online algebra ...
1
vote
1answer
41 views

$A$ is diagonalizable, then exists $B^3=A$

I got this question on my linear algebra exam. The question was to prove/disprove the following statement: If $A$ is a diagonalizable matrix. Then there exists a matrix $B$ so that $B^3 = A$. I ...
0
votes
0answers
47 views

Meaning of $\alpha^\beta$ when $\alpha, \beta \in F_2[x]/g(x)$? [closed]

Let $F$ be a finite field extension $F_{2^m}[x]$ (mod $g(x)$) where g(x) is an irreducible polynomial of degree $m$. Let $\alpha, \beta \in F$. What (if anything) is $\alpha^\beta$? I've read in ...
0
votes
0answers
26 views

How to calculate modulus of remainder when a large number is divided by other.

The problem is : I have to calculate the remainder for a very large number but if answer overflow, divides the result by $10^9+7$. Let $a,b,c$ are numbers such that: $1\le a,b,c\le 10^{12}$. How to ...
0
votes
1answer
43 views

Is it possible to graph $f(x) = (-2)^x$?

I can plot points on the coordinate plane only when my x value is integer. What's happening on the graph when x is fraction? My ...
0
votes
0answers
77 views

GCD of exponents

I have been trying to determine a shorter method for calculating $\gcd(a^n + b^n, |a-b|).$ I noticed that, if I were to calculate (using the above formula) say $a = 100$ and $b = 4,$ starting from $1$ ...
0
votes
3answers
56 views

Estimating the value of $\ln2$ using $e^3$ and $2^{10}$

I found this question in an old MAT paper but I'm not getting very far. You are given that $e^3$ is approximately $20$ and that $2^{10}$ is approximately $1000$. Using this information a student can ...
6
votes
2answers
101 views

Can you prove the power rule for irrational exponents without invoking $e$?

The power rule states that for any real number $r$, $$\frac{d}{dx}x^r=rx^{r-1}$$ Now one common way to prove this is to use the definition $x^r=e^{r\ln x}$, where $e^x$ is defined as the inverse ...
2
votes
3answers
46 views

A Question About Square Roots And Exponent Laws

Why is it in math, $\sqrt{ab}$=$\sqrt{a}$$\sqrt{b}$? I get why this is the case for any other power instead of $1/2$. For instance, if the power was for, then $(ab)^4$=$(a)^4$$(b)^4$ because on both ...
0
votes
1answer
32 views

Solving $KL^{2C} - LC^{2C} = (B-C)C^{2C} - C^{2C}(T+KG^2) - KP^2$ for $L$

How can I solve this equation for $L$. Everything else is known number: $$KL^{2C} - LC^{2C} = (B-C)C^{2C} - C^{2C}(T+KG^2) - KP^2$$
0
votes
1answer
78 views

Asymptotics to $f(n) = \int_0^1\bigl ( \frac{ \operatorname{li}(x)}{x} \bigr)^{2n + 1} \,(x-1) \, dx $

Consider $$f(n)=\int_0^1\Bigl(\frac{\operatorname{li}(x)}{x}\Bigr)^{2n + 1}\,(x-1)\,dx $$ Where $n$ is a positive integer. (I know that $f(1) = \zeta(3) $ but I already made a Question about ...
0
votes
0answers
19 views

Separate multiplication of element-wise exponential

If I have an equation of the form $A*(B*x).^4$ where $A$ and $B$ are rectangular matrices of size $n\times n$ and $x$ is a vector size $n\times 1$. I would like to get $B$ out of the brackets, but ...
0
votes
0answers
25 views

Factoring a number in the format of 2^x + 2^y … ect

Lets say I have this number 4294967296, I can easily convert it to 2^32. Now lets say I have this number 33554688, How could I factor it to 2^25 + 2^8. It needs to always be in the format of (2 to ...
1
vote
0answers
100 views

Exponential decay of the function

Let $f$ be a continuous function on $S^2$. Consider $g\in C^{\infty}(R)$, such that $g(x)=1$ for $|x|\leq 1$ and for $|x|\geq 2$. Let $h(x)=g(x)-g(2x)$. The notation $proj_k$ denotes the orthogonal ...
12
votes
6answers
2k views

Between any two powers of $5$ there are either two or three powers of $2$

Is this statement true? Between any two consecutive powers of $5$, there are either two or three powers of $2$. I can see that this statement is true for cases like $$5^1 < 2^3 < 2^4 < ...
2
votes
2answers
102 views

What's the formula for producing these series? [closed]

I have a function which produces a series from an integer. I currently do this iteratively. ...
1
vote
1answer
71 views

How to find the least positive $K$ such that $N^K \equiv 1 \pmod{P}$ where $P$ is prime and $P$ doesn't divide $N$?

I noticed that this $K$ is one of the divisors of $P-1$. So my solutions is looping on the divisors of $P-1$ in ascending order, till I find the first divisor $d$ where $N^d \equiv 1 \pmod{P}$. ...
1
vote
3answers
146 views

What is wrong with the reasoning in $(-1)^ \frac{2}{4} = \sqrt[4]{(-1)^2} = \sqrt[4]{1} = 1$ and $(-1)^ \frac{2}{4} = (-1)^ \frac{1}{2} = i$?

$$(-1)^ \frac{2}{4} = \sqrt[4]{(-1)^2} = \sqrt[4]{1} = 1$$ $$(-1)^ \frac{2}{4} = (-1)^ \frac{1}{2} = i$$ Came across an interesting Y11 question that made pose this one to my self. I can't for the ...
2
votes
1answer
56 views

The concept of an real irrational power [duplicate]

One can understand the concept of natural power, as $x^n$, being a product of a number by itself $n$ times: $$x^n=\underbrace{x\cdot x\cdot\dotsb\cdot x}_n$$ We can also get the idea of a rational ...
1
vote
1answer
15 views

Isolating Exponent Variables for Deriving Equations

Our Math class right now is all about Geometric Sequences and we all know the equation for them which is $$A_n = A_1 \cdot r^{n-1}$$ We were only given this particular equation so to make things ...
2
votes
1answer
49 views

Is $(2^2-1)(3^2-1)(4^2-1)…(300^2-1)$ divisible by $7^{95}$?

Is $(2^2-1)(3^2-1)(4^2-1)....(300^2-1)$ divisible by $7^{95}$? And what about $7^{100}$? This is taken out of one of the TAU entry tests. I seem to always have a struggle with these exercises, ...
2
votes
0answers
53 views

Where can I find the proofs of the properties of exponentiation when powers are members of ℕ, ℤ, ℚ, ℙ, and ℝ?

Where can I find the proofs of the properties of exponentiation when powers are members of ℕ, ℤ, ℚ, ℙ, and ℝ? So far, I have only found this: http://andrusia.com/math/preliminaries/...
0
votes
1answer
34 views

Is it the same $a^\alpha + b^\alpha = x$ as $(a^\alpha)^\frac{1}{\alpha} + (b^\alpha)^\frac{1}{\alpha} = x^\frac{1}{\alpha}$

I want to ask if it is possible to transform equation $a^\alpha + b^\alpha = x$ into $a + b = x^\frac{1}{\alpha}$ by elevating each parameter by $\frac{1}{\alpha}$? Or if we elevate each parameter by $...
1
vote
2answers
53 views

A simple question on modular arithmetic and exponentiation [closed]

I am trying to understand modular exponentiation and its implementation in C from here On my way of understanding it I came across the following equation: $$(x^{n/2}\bmod M)^2\equiv(x^2\bmod M)^{n/2}$...
1
vote
2answers
48 views

Comparing powers

Is there any way to check besides acctually calculating whether $n^x$ is $> = <$ then $m^y$? For example how to check wheter $440902^{532446} > = < 555151^{523163}$?
0
votes
1answer
38 views

simplification of (i^j)%k

I'm writing a program that is spending major amounts of time of CPU doing power operations. I need to get the modulo of $i^j$ by k. Is there any simplification of $i^j$%k that doesn't involve an ...
0
votes
2answers
49 views

Strange Logarithm Phenomenon

I was looking at the following two equations and was trying to solve them when I noticed a very strange phenomenon. These were the equations I was trying to solve: $y=\log_3 x$ & $y=\log_x 3$. I ...
0
votes
2answers
55 views

Is mod of exponent allowed?

Lately I was studying about Modular Arithmetic and the way modulus is used to calculate large numbers. What caught my attention was calculating powers with modulus. It's generally that we calculate ...
0
votes
2answers
53 views

Instead of $x^{4/3}$ do $(x^4)^{1/3}$

I'm doing a little filter processing research and at one point I expand a signal by applying: $x^{4/3}$ to the signal which will only yield real outputs for real, positive inputs. Since $(x^a)^b = (x^...
-2
votes
1answer
29 views

Convention for Expressions Involving Exponents

Is there a reason why positive exponents are preferred in some settings over negative? Further, I've noticed if there's a positive rational exponent, then it is sometimes expressed as a product with ...
0
votes
1answer
32 views

can we preserve correctness of inequality after adjusting all involved exponents?

Problem Lets assume we have an inequality such that it involves only positive, real values and all exponents within this inequality are integer multiples of $2n$ (even), we suppose this inequality is ...
3
votes
3answers
751 views

The Exponential Property

Prove true or false for the statement: every $x \in \mathbb{R}$, holds $x^{\frac{6}{2}} = x^3$ The habit of what we did everyday when facing exponential forms like this creates confusion to prove ...
0
votes
1answer
37 views

Modular Exponentiation with power not within long long limit.

I came across a problem where I need to find $x^y$ mod $p$ where $p$ is prime.It is an easy problem which can be find in $O(\log y)$ complexity but the twist in the problem is that value of $y$ is $n\...
2
votes
1answer
46 views

Prove that for any $g\in G$ and $m,n\in \mathbb{Z}$, $g^{m}g^{n}=g^{m+n}$.

Can someone please tell me if my solution is okay? Prove that for any $g\in G$ and $m,n\in \mathbb{Z}$, $g^{m}g^{n}=g^{m+n}$. Case 1: $m,n>0$ $g^{m}g^{n}=g\cdot g\cdot \cdots \cdot g$ ($m$ $g$'...
0
votes
0answers
115 views

Fastest way to calculate $a^\binom{n}{r}\bmod M$ if $\binom{n}{r}$ is very large

What is the fastest way to calculate $a^\binom{n}{r} \bmod M$? Here $M$ is a prime number. So far the best approach I can get is to use $\binom{n}{r}=\binom{n-1}{r-1} + \binom{n-1}{r}$ and then apply ...
0
votes
2answers
32 views

$x^2 * (x+1)^2 = (x^2+x)^2$|| I do not understand this

$$x^4(x+1)^4(2x+1)$$ I was solving this question (I had integrate it) but I did not know how to solve it! So I saw the solution and it says $$x^4(x+1)^4 = (x^2+x)^4$$ and I do not get how that works?...
3
votes
2answers
243 views

Irrational exponent understanding [duplicate]

For an integer number $a$ $$x^a=\{(x)(x)(x)...(x)\} (a\,times)$$ $$x^{\frac{1}{b}}=n\rightarrow\;\{(n)(n)(n)...(n)\}(b\,times)=x$$ For rational number $m=\frac{a}{b}$ $$x^m=x^\frac{a}{b}=(x^a)^\...
1
vote
0answers
27 views

if ${m/n}$ is an approximation to $\sqrt{2}$, prove $m/2n+n/m$ is always a better approximation. [duplicate]

I noticed $(m/2n+n/m)^2 -2$ = $(m/2n-n/m)^2$. So I set $(m/n)^2 - 2 = d$ and substituted value for n in the above equation but can't show that resulting equation is less than d.
-3
votes
1answer
34 views

how to calculate $(b^{(p \bmod m)}) \bmod m$? [closed]

How to calculate a base to the power an exponent which is extremely large and is already modded with a prime number m and in turn the whole expression is also modded with the same m.
0
votes
0answers
41 views

How to convert my equation (Exponential) to cot form?

I have the below equation: $$ F(t)=(k_1 e^{at} + (k_{Re}-ik_{Im})(\lambda+i\omega)e^{(\lambda+i\omega)t}+(k_{Re}+ik_{Im})(\lambda-i\omega)e^{(\lambda-i\omega)t})/((r_1k_1 e^{at} + r_2(k_{Re}-ik_{Im})(...
-2
votes
1answer
49 views

Let $a$ and $b$ be real numbers such that $a>b , 2^a +2^b=75$ and $2^{-a} + 2^{-b} =1/12$ , find the value of $2^{a-b+2}$

I have tried till this how to solve further please help. $$75/12=(2^a+2^b)(2^{-a} + 2^{-b}) =2+2^{a-b}+2^{b-a}$$
1
vote
2answers
37 views

The logic in radical symplification

I'm having troubles while studying radicals, namely with converting expressions with the form $$\sqrt{a+b\sqrt{c}}$$ to $$a+b\sqrt{c}$$ and vice versa. When I'm dealing with these kind of problem, I ...
0
votes
1answer
27 views

What is the value of $x$ when $a b^x$ > $ c d^x$

I am trying to find the min value of x, which make that inequality true $a b^x$ > $ c d^x$. And is there any cheat sheet that contains all the formulas to solve those type of inequality and ...