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

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0answers
26 views

Rational exponentiation?

Consider the following operation: $\left(\frac{a}{b}\right)^\frac{n}{m}$ where $a, n\in\mathbb{Z}$ and $b, m\in\mathbb{N^*}$. My question is: when the result is a rational number, how (formula or ...
2
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2answers
286 views

How to compute $2^{\text{some huge power}}$

I have to compute $$2^{p-1} \mod p$$ and show by Fermat's little theorem that $p$ isn't prime. I know what the question is asking but I'm not sure how to reduce the exponent on $2^{p-1}$ to a more ...
2
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5answers
516 views

How to quickly identify perfect powers

In a test I'll take there may be a question such as the following: A perfect power is an integer that can be written as $a^b$, $a$ and $b$ being integers greater or equal to 2. One of the ...
2
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1answer
163 views

Trouble solving polynomial equation with exponent

I'm having trouble solving this equation.It looks simple, but I just can't find the answer.Can someone help me? $$9x^4-13x^2+4 = 0$$
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1answer
129 views

Is $2^n \mod m \equiv (2^{n/2} \pmod m ) ^ 2 \pmod m$?

I'm trying to write a procedure that solves (2^n - 1) mod 1000000007 for a given n. n can ...
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2answers
282 views

What does $i^i $ equal and why? [duplicate]

I've been reading up on why the value of 0^0 is controversial (see Zero to the zero power - Is $0^0=1$?) and I wondered: is it possible for $i^i$ to have a value? I plugged it into a TI-83 calculator ...
5
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2answers
55 views

Upper bound for product of exponents

From here we have the bound $$\left(1-\frac1N\right)^N\leq e^{-1}$$ where $N$ is a positive integer. Written another way, it is ...
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1answer
31 views

Question about basic exponential/logarithm properties

Solve for $k$: $$e^{k/2}=a$$ Solution: $$e^{2k}=a$$ $$ k/2 = \mathbf{ln}a$$ $$ k=2\mathbf{ln}a$$ $$= \mathbf{ln}a^2$$ My question is: why does $2\mathbf{ln}a = \mathbf{ln}a^2$? Why can you ...
0
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0answers
116 views

Development of imaginary exponent without appealing to “ambiguity” between $i$ and $-i$

Is there a way to develop the definition of the imaginary exponent, $z^i$, for complex $z$, that does not appeal to the notion that $i$ and $-i$ are "qualitatively indistinct" and that does not rely ...
1
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0answers
35 views

Math equations of electron scattering

I'm trying to figure out the missing step here, in a problem about X-ray crystallography. I am referring to the attached image: In the image, A= electron density, Z= distance traveled, λ= X-ray ...
0
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1answer
92 views

Unary minus on squared number

According to algebra as I know it, $-2^2 = 4$, but most calculators expand this to $-2 * 2 = -4$, which yields a different answer. This is because of the order of precedence. In traditional math, ...
3
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1answer
99 views

find value (-2)^-(2)^(-2)

Find the value of $(-2)^{-(2)^{(-2)}}$. Is it 16/8/-8/none? My attempt: $a^{-x}=\frac1{a^x}$, so, $(-2)^{-(2)^{(-2)}}=(-2)^{\frac{-1}{2^2}}=\frac{1}{(-2)^{\frac14}}$. That is, I would pick 'none ...
5
votes
4answers
125 views

Convergence of exponential matrix sum

Let $A$ be an $n\times n$ matrix. Consider the infinite sum $$B=\sum_{k=1}^\infty\frac{A^kt^k}{k!}$$ Each term $\dfrac{A^kt^k}{k!}$ is an $n\times n$ matrix. Does the sum $B$ always converge? (i.e. ...
0
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1answer
51 views

Modular Exponentiation Equivalence Problem

Find the integer $a$ such that $0 \leq a < 113$ and $102^{70} + 1 \equiv a^{37} \bmod{113}$. I started off by using modular exponentiation to realize that the left side of the congruence is ...
2
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3answers
64 views

computing $2^{170}+ 3^{63}\pmod {19}, 3^{175} + 2^{73} \pmod {17}$, etc… by hand

I came across several questions like this in the problem section of a book on coding theory & cryptography and I have no idea how to tackle them. There must be a certain trick that allows for ...
1
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0answers
95 views

Double summation of elementary functions

I am finding some trouble on calculating the following double summation: $ \sum_{k=1}^\infty \frac{b^k}{k!k}\sum_{n=0}^{k-1}\frac{(b*x)^n}{(n-1)!} $ Note that the inside sum gives: ...
0
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1answer
34 views

Equation with a division in exponent

I have an equation: 10,9 * 2^(x/1,5) = 1000 and want to calculate the value of x. x being in the exponent is my problem. How can I get to something like: x = ...
3
votes
4answers
362 views

Raising a Complex Number to a Decimal Value

So for my class i have to make a java program that deals with complex numbers. I finished getting the root and power and i was wondering how to do a method that deals with powers such as 2.56. Now im ...
0
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1answer
140 views

big-O proof with power functions

I was wondering if anyone could show a proof for why $a^x$ is $\mathcal{O}(b^x)$ if $a$ and $b$ are constants and $a < b$. In other words, with power functions, does the function with the largest ...
3
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3answers
178 views

$x^{y^z}$: is it $x^{(y^z)}$ or $(x^y)^z$?

Of the following, why is a usually considered true, and for what reason other than "tradition" and "more convenient"? a: ${x}^{y^z} = x^{(y^z)} \neq {(x^y)}^z$ b: ${x}^{y^z} = {(x^y)}^z \neq ...
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1answer
106 views

How to solve $5^n - 5^{n-3} = 5^{n-3} *124$

how is $$5^n - 5^{n-3} = 5^{n-3} *124$$ Can anybody provide a step by step solution.I will greatly appreciate if any online source for such material is provided. Regards
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2answers
260 views

x raised by the power of y equals infinity.

How many zeros are there in this equation? 10000^999 In my calculator it says infinity but that doesn't seem right.
0
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2answers
140 views

Termination of a Fast Exponentiation problem

Here's the problem I am stuck on. There exists a fast exponentiation program like the following: Given inputs a in the set of all Real numbers, b in the set of Natural numbers, initialize ...
4
votes
3answers
255 views

$\log(n)$ is what power of $n$?

Sorry about asking such an elementary question, but I have been wondering about this exact definition for a while. What power of $n$ is $\log(n)$. I know that it is $n^\epsilon$ for a very small ...
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2answers
68 views

List the numbers in order

How would I list these numbers in order without using a calculator? Thank you List these numbers in increasing order: $2^{800}$, $3^{600}$, $5^{400}$, $6^{200}$
2
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1answer
61 views

What is the exponential for the matrix

What is the exponential for the matrix $$ \begin{pmatrix} 0 & -x & 0 \\ x & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix} $$ Is it $$ ...
1
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2answers
28 views

On the definition of product of groups

What I'm asking comes from Bosch, Algebra, the first chapter on elementary group theory. 1) Let $X$ be a set and $G$ a group. Then the set $G^X$ of maps from $X$ to $G$ is a group in a natural way, ...
0
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1answer
31 views

For each of series find the smallest $k$, that $a_n = O(n^k)$

Hey I need you to check my solutions: a) $a_n = (2n^{81.2}+3n^{45.1})/(4n^{23.3}+5n^{11.3})$ This one is done from $\sum_{i=1}^{k} O(a_i(n)) = O(max\lbrace a_i,..,a_k \rbrace )$ So it's ...
2
votes
1answer
196 views

Modulo operation of large powers

I came through this property in a cryptography book. $(ab)\bmod n=\bigl((a \bmod n)(b \bmod n)\bigr)\bmod n$. There is an example in the book, $10^n\bmod 3= (10\bmod n)^n$. Now if I have ...
0
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4answers
81 views

Method to solve this type of equation

It seems simple yet I have to show how I got to the answer... I've been Googling, but can't narrow it down. $8 = 2^x$ Thanks in advance.
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1answer
162 views

Solve $a=x^n$ , $b=(x+1)^n$ for $x,n$

$$a=x^n~,~b=(x+1)^n$$ Just trying to solve these for $x$ and $n$ . For some reason WolframAlpha gives me a blank screen? Much thanks for any help.
0
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1answer
32 views

How to evaluate $\pm$ operations

When finding the root of a number with an even exponent, $x^y$ becomes $\pm x$. How would this work in a situation such as $a = \sqrt{(5x + 12)^2 + m}$? I know that the result is not $a = \pm 5x + ...
0
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3answers
900 views

Something to the power of a logarithm

This is probably a very obvious question, but here goes... An answer in my textbook claims that $$3^{\log n} = n^{\log 3}$$ and that $$4n^2 (3/4)^{\log n} = 4n^{\log 3}$$ Why, using more basic ...
7
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1answer
162 views

About $e^{\pi}\gt {\pi}^e, \ e^{e^{\pi}}\lt {\pi}^{{\pi}^{e}},e^{{\pi}^{{\pi}^e}}\gt {\pi}^{e^{e^{\pi}}}$ and their generalization

Let us define a sequence $\{(a_n,b_n)\}$ as $$(a_1,b_1)=(e,\pi),\ \ (a_{2n},b_{2n})=(e^{b_{2n-1}},{\pi}^{a_{2n-1}}),\ \ (a_{2n+1},b_{2n+1})=(e^{a_{2n}},{\pi}^{b_{2n}})$$ for $n=1,2,3,\cdots$. Then, ...
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0answers
59 views

Algorithm for finding power

I has been searching for a high precision library in PHP to do calculations like $$232323232323^{121212.2232323232}$$ etc (ie, with very large numbers, including decimals), but failed to get any. ...
3
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4answers
433 views

How to explain Fractional and Negative Exponents

My classmates doesn't understand Fractional and Negative exponents, since I was the top of my class, so they all came to me... Is there any way to explain it clearly to them?
0
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2answers
68 views

Simplifying euler exponent?

How would I go about simplifying and finding the exact value for this question: $e^{6\ln(4)}$ I know that $e ^{\ln x} = x$ but how does the $6$ affect this answer?
1
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2answers
76 views

Math question from the GMATprep

If $xy=1$ what is the value of: $2^{(x+y)^2}/2^{(x-y)^2}$ A 1 B 2 C 4 D 16 E 19 $(x+y)^2/(x-y)^2$ because $2$ just cancels out from numerator and denominator, ...
4
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4answers
205 views

Last three digits of 23^320

What is the best way to compute the last three digits of $23^{320}$? I know one way is by starting of $23^2$ and finding the last three digits, then squaring those (calculating $23^4 \pmod {1000}$) ...
2
votes
1answer
174 views

Solve a system of equations with $3$ unknown powers?

I'm trying to solve this, knowing $X$ $$\begin{cases}1^x*2^y*3^z=X\\x+y+z=13\end{cases}$$ So for example, if $X=2048$ , we have $$x=2\\y=11\\z=0$$ I barely have memories from high school ...
47
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2answers
2k views

Fastest way to check if $x^y > y^x$?

What is the fastest way to check if $x^y > y^x$ if I were writing a computer program to do that? The issue is that $x$ and $y$ can be very large.
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1answer
73 views

Formula for the number of digits in the number $2^x$

I'm wondering if there is a formula for the number of digits in $2^x$. For example if $x = 3$ then the number of digits is equal to $1$ because $2^3 = 8$ or for example if $x = 4$ then the number of ...
2
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1answer
434 views

The domain of fractional exponents

Take the following: $$f(x) = x^{6/4}$$ The domain of this function is all real numbers. This function can be simplified to: $$f(x) = x^{3/2}$$ The domain of this function is all real numbers ...
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2answers
131 views

Checking inequality without actually calculating LHS and RHS

How to check whether the following inequality is true or not without actually calculating the values of $x^y $ and $y^x $: $$ x^y > y^x$$ (x and y are integers)
3
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3answers
146 views

Can a fourth-order equation be solved like a quadratic equation?

I was asked to find the zeros of $y = x^4 + 5x^2 +6$. I tried to turn this into a quadratic to factor it as follows: $y = x^4 + 5x^2 +6 = {(x^2)}^2 + 5{(x^2)}^1 + 6$ Put another way: Let $t = ...
0
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2answers
164 views

Rational exponents: prove some states

In some rational exponent expressions the solution isn't a real number why? Example (explain what I mean): $$\begin{align} \Big(-x\Big)^{1/n}=\left\{\text{is not a real number}\right\} \end{align}$$ ...
1
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1answer
48 views

Is the statement $a+b < c+ d \implies e^{-a} + e^{-b} > e^{-c} + e^{-d}$ true?

As the title says, I am trying to ascertain whether the following is true: Suppose $a,b,c,d\in \mathbb{R}^+$ are such that $a + b < c + d$, then it is also true that $e^{-a} + e^{-b} > e^{-c} + ...
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2answers
74 views

Why is $0^0=1$, given the following information? [duplicate]

Why is $0^0=1$, given the following information? We really have two separate rules that are at odds with each other. Typically we have $0^n=0$ (provided n is positive) and $a^0=1$. Each of these ...
0
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2answers
97 views

How to find the remainder of $(2010^{1020} + 1020^{2010})$ divided by $3$

What is the remainder when $2010^{1020} + 1020^{2010}$ is divided by 3?
2
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4answers
231 views

How to find $\sqrt{1+{4\over x}+{4\over x^2} }$?

If $$abx^2 = (a-b)^2(x+1)$$ then what is $$\sqrt{1+{4\over x}+{4\over x^2} }$$ (A) $a+b \over a-b$ (B)$a-b\over a+b$ (C) $a^2+ab$ (d) None EDIT: What I've done is this: ...