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

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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
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1answer
219 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 ...
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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
183 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
33 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
1k 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
169 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, ...
1
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0answers
63 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. ...
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4answers
484 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?
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2answers
70 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
211 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
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1answer
204 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 ...
49
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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.
1
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1answer
75 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
484 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 ...
1
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2answers
140 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
161 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
175 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
49 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
76 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
108 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
236 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: ...
1
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1answer
100 views

How to solve this equation with parameters on power?

Please let me know how to solve this equation: $$100^{1-b}=\frac{1}{2}40^{1-b}+\frac{1}{2}200^{1-b}$$ I try to use the trick of $x=e^{\log x}$ But it doesn't work And $b\not=1$
8
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1answer
174 views

Need a general formula for $\frac{d^n}{dx^n}\left(f(x)^m\right)$

Let $m,n\in\mathbb{N}$. I need to express the derivative $\displaystyle\frac{d^n}{dx^n}\left(f(x)^m\right)$ in terms of sums/products of the derivatives of the function $f$ itself. Here are results ...
1
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2answers
36 views

threshold of n to satisfy $a^n <n^a$

How to find the minimum of $n$ when we know $a$, to satisfy: $a^n<n^a$ $a^m>m^a$ for each $m>n$ $n$ and $m$ are natural numbers.
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1answer
46 views

Calculate the Burning Time for a Lamp

If you have a lamp with burning time 4000 hours. If the time goes forward until the lamp will be destroyed the exponential distribution is 3675 hours, what is the probability of a lamp to be working ...
2
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1answer
65 views

How to find $x + y + z$?

Q. If $x^{1/3} + y^{1/3} + z^{1/3} = 0$, then (A) $x + y + z = 3 xyz$ (B) $x + y + z = 0 $ (C) $( x + y + z)^3= 27 xyz$ (D)$ x^3 + y^3 + z^3 = 0$ What I've done: ...
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0answers
1k views

How to solve polynomial-exponential equation

I'm trying to solve equations like the following one: $$5 + 3x - 4x^3 = e^{x^2}$$ I've tried using the Lambert W function, but I didn't get any success. I must admit I'm relatively new to Lambert W ...
5
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1answer
76 views

Can this type of limit even be evaluated?

This is going to be a long question; please bear with me. We are familiar with the notations $\Sigma^{n}_{k=0} a_k$ and $\Pi^{n}_{k=0}a_k$ for the sum and product of the finite sequence $\{a_n\}$. ...
1
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2answers
243 views

Exponential equation with absolute value: $9^{|3x-1|}=3^{8x-2}$

$$9^{|3x-1|}=3^{8x-2}$$ Can someone show me the steps on how to solve this, i've been trying for 30 minutes
5
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2answers
242 views

Fraction raised to integer power

if I have $(p/q)^n$ where $p,q,n$ are integers and $p/q$ is a... I don't know what you call it. Not a whole number, but something like 15/7 where you can't reduce it any more and it's non-integer. Can ...
2
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0answers
229 views

Solving $x$ for $y = x^x$ using a normal scientific calculator (no native Lambert W function)?

Solving $x$ for $y = x^x$ using Lambert W function is clear enough thanks to this handy answer, but as I'm using the solution in a network support document I need it in a form that can be solved on a ...
0
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1answer
63 views

Relation of $e$ to other numbers…

I found the following result, When i was working on my calculator . $$x^y < y^x \quad ,x < y \quad \text{ for } x,y<e$$ $$x^y > y^x \quad ,x < y \quad \text{ for } x,y>e$$ I can't ...
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1answer
62 views

How do I simplify this expression?

How do I simplify this expression? $$\frac{(4-x)^2 (1/3) (6x+1)^{-2/3} (6) - (6x+1)^{1/3} (-2x)}{(4-x^2)^2}$$
0
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1answer
249 views

How do I simplify the expression (a^-1 + b^-1) ^-1?

How do I simplify the expression ... $$(a^{-1} + b^{-1}) ^{-1}$$ ?
0
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1answer
44 views

Simplifying exponentials of the form $\,a^x \cdot b^y$

I am given the exponential $\left(\dfrac{1}{2}\right)^x\cdot 4^{(x/2)}$. While my intuition screams that this can be simplified to $\dfrac{2^x}{2^x} = 1$, I am unable to see a concrete mathematical ...
1
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0answers
50 views

Matrix exponent and representations of $\mathbb{R}$

It is well known that $\exp(C^{-1}AC)=C^{-1}\exp(A)C$, where $C$ is an invertible matrix. Really, $$\exp(A)=\sum{}\frac{A^k}{k!} \quad \text{and} \quad (C^{-1}AC)^k=C^{-1}A^kC,$$ so the first ...
0
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2answers
41 views

Simplify the expression.

Simplify the following expressions using fractional exponents, I forgot how to do this type, do I rationalize it? or can it just cancel each other out? $ {\Large \frac{\sqrt[ 5 ]{x^{ 3 }}}{ \sqrt{x} ...
0
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1answer
163 views

Simplify the following expressions using fractional exponents

Simplify the following expressions using fractional exponents. Display your answer using fractional exponents. $${ \sqrt[ 3 ]{x^{ 7 }} = }\text{ and }{ \sqrt[ 7 ]{x^{ 3 }} = }$$ Thanks..not sure ...
0
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1answer
61 views

What's the pattern? roots and powers of n?

I won't hold it against anyone if this is considered a bad question but I just don't know how else to put it and really want to know. 1st case: n = positive integer a = $\sqrt n$ b = 1 2nd case: n = ...
2
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0answers
93 views

Is $x^{\frac 12}$ the same as sqrt(x)

maybe this question is very simple and clear and trivial to everbody. but right now i'm not sure. the equotation $$ x^\frac{1}{2} = \sqrt{x} $$ is only true whenever $ x \geq 0 $ right? the square ...
8
votes
1answer
187 views

Expressing the values of a matrix at pow N

I have a square matrix (that comes from a Markov Chain) that looks like that: $$Q = \begin{bmatrix} 0 & 1& 0 & 0 & .. & 0 & 0\\ 0 & a & 1-a & 0 & .. & 0 ...
1
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1answer
87 views

Systems of equations question

\begin{align*}a^a\cdot b^b\cdot c^c\cdot d^d&=\frac12\\a+b+c+d&=1\end{align*} How can we find solutions for this system of equations given that $a, b, c, d > 0$ ?
2
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2answers
232 views

A non-exponentially bounded analytic function?

A function $f:\mathbb R\to\mathbb R$ is said to be exponentially bounded if there is an $n$ such that for sufficiently large $x\in\mathbb R$, $\exp(\exp(\cdots \exp(x)))>f(x)$ (where the $\exp$ is ...
10
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1answer
1k views

What is wrong with this funny proof that 2 = 4 using infinite exponentiation?

Out of boredom, I decided to recall the following equation: $$x^{x^{x\cdots}} = 2.$$ Which, I simply rewrote like this: $x^2 = 2$, and therefore $x = \sqrt{2}$. Then I took a look at the more ...
2
votes
2answers
48 views

$P[ e^{tX} > e^{ta} ] =?$

Can anyone help me understand why given any random variable $X$, the following stands true? $$ \forall t > 0, P( e^{tX} > e^{t\epsilon} ) \le e^{-t\epsilon} E[e^{tX}]. $$ I found it in the ...
2
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4answers
197 views

Exponent rules with negative numbers

Hello if you could answer this please what is the difference between -2^2 and (-2)^2
1
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1answer
105 views

How to solve strange exponential equation?

How would equations of the form $b^x-x^a=0$ be solved for $x$, given $a$ and $b$? For instance, specifically, how would $2^x=x^2$ be solved? Does a method exist?