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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1answer
19 views

Properties of the exponent function attached to a nonzero prime ideal in a Dedekind domain

I want to prove properties of $v_\mathfrak{p}$, which I have been told is: "the exponent function attached to a nonzero prime ideal $\mathfrak{p}$ that maps a given nonzero fractional ideal to the ...
2
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0answers
29 views

A relation with limits

Let $t \in \mathbb{R}_+$, $\varepsilon > 0$ and $p_\varepsilon \in [0, 1]$. There is in a book the following relation: $$ \underset{\varepsilon \rightarrow 0}{lim} (1 - p_\varepsilon)^{\lceil t / ...
0
votes
1answer
174 views

why does e raised to the power of negative infinity equal 0?

Why is it that e raised to the power of negative infinity would equal 0 instead of negative infinity? I am working on problems with regards to limits of integration, specifically improper integrals ...
1
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3answers
38 views

Operations and Identities [duplicate]

We have the binary operation addition on numbers. It has an additive identity ( 0 ) and it is commutative. Multiplication is simply repeated addition. It is a binary operation on numbers. Its ...
1
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4answers
46 views

How to see that $2^{n-1} + 2^{n-1} - 1 = 2^n - 1$

How to see that $2^{n-1} + 2^{n-1} - 1 = 2^n - 1$? Is there a rule about adding two powers of the same base I'm not aware of? I know that you can "add the exponents" if you are multiplying numbers ...
1
vote
1answer
17 views

Proof that. for any two natural numbers $n$, $m$, $n^m$ is not an $n$-ary pernicious number.

$a_{10}$ is defined to be an $n$-ary pernicious number when the digit sum of $a_n$ is prime in base $10$. How can I prove that, for any two natural numbers $n$, $m$, $n^m$ is not an $n$-ary ...
0
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2answers
29 views

Implicit logarithmic differentiation to find the horizontal tangents of an exponential function

The graph of $y = 6{(3{x}^2)}^x$ has two horizontal tangent lines. Find equations for both of them. $$ \\ \begin{align} \\ y &= 6{(3{x}^2)}^x \\ y &= 6 \cdot {3}^x \cdot {x}^{2x} \\ \ln{y} ...
0
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1answer
34 views

Variable exponent solve for x

How can I solve this exponent problem using simple math only? We need to solve for $x$ $2^{2x}-(3.2)^{x+2} + 32 = 0$ The second term here is $3.2$ not $3\cdot2$ ie 3 decimal 2 not 3 into 2. My ...
1
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0answers
28 views

Simplified Exponent in $\bmod$ equation

I am trying to simplified the following expression: $$\left(y^m\right)^{x} \bmod p$$ In my case, I can only solve $(y^x) \bmod p$ first without prior knowledge of $m$. Eventually, my answer should ...
0
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2answers
38 views

Is it wrong to “imagine” a one when thinking about exponentiation? (e.g. $3^2 = 1 \times 3 \times 3$)

This might be a bit of a basic question, but I'm going through Khan Academy to refresh my math skills in order to pursue a self-study of higher mathematics, so I'm really focused on the "why" of the ...
0
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1answer
23 views

Multiplication Question with Powers

A quick question. I dont know how to do this. $$1-2^k\times 2+1\times (-1)$$
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0answers
9 views

Find distortion exponent from Fourier fitting

I'm facing this problem in my master thesis: we are measuring the signal from a sensor which is, physically, a $\sin^2$ (or $\cos^2$). Some non idealities distort the signal by introducing an exponent ...
1
vote
1answer
65 views

Besides $2^4$ and $4^2$, are there any other numbers that, with the base and exponent flipped, will equal the same value? [duplicate]

I've noticed that when you flip the base and the exponent in $2^4$ to get $4^2$, you get the same value, $16$. If there are any other numbers that can make this work, let me know. This is just ...
0
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0answers
25 views

exponent and modulus

Good day all, I am working on a equation and not getting it to work correctly. Hope to seek some advice. $r1 = (m+xr-m1)x^{-1}\pmod q$ $y^{r1} = (y^m)^{x^{-1}} \times y^{xr(x^{-1})} \div ...
1
vote
1answer
31 views

How can this exponential equation be computed

Is there a mathematical way to solve such equations, besides try and error of course? $e^{-x} = 1-x/5$
1
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1answer
56 views

Proving something about the sequence of powers of 3 mod 10. Oh boy

Given the sequence $t_0=3, t_1=3^3,...$ so that $t_{n+1}=3^{t_n}$, prove $t_{k+1} \equiv t_k \mod 10^n$ for all integers $n \leq k$ My work so far: I thought it was a pretty obvious case for ...
1
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3answers
21 views

Explanation for an equality involving complexed exponents

$$\frac{e^{i(N+1)x}-e^{-iNx}}{e^{ix}-1} = \frac{e^{i(N+1/2)x}-e^{-i(N+1/2)x}}{e^{ix/2}-e^{-ix/2}}$$ I'd be glad to get an explanation for both numerator and denominator. Thanks in advance!
3
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4answers
92 views

Find derivative of $x^{x^x}$

Trying to find the derivative of: $$ x^{x^x} $$ I have a solution but cannot understand the third transition:
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3answers
76 views

Question regarding the square root of a squared number. [duplicate]

I've learnt that the square root of a number squared is equal to the absolute value of that number, but I haven't really understood why. I have looked through other questions on MSE but didn't really ...
1
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2answers
53 views

Prove that 100…500…1 (100 zeros in each group) is not a perfect cube?

How can i prove that 100...500...1 [100 zeros in each group ( ... is 100 zeros)]is not a perfect cube? I tried symmetric features of the number but could not figure out anything related.any ideas ...
3
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1answer
82 views

Arrangement of integers in a row such that the sum of every two adjacent numbers is a perfect square.

Inspired by this interesting question and in order to solve an old problem, I have the following question: Can we construct a strictly increasing sequence $(N_i)_{i\in \mathbb{N}}$, such that for ...
0
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0answers
17 views

A more general class for power-law distribution

Is a power-law distribution, also a: Fat-tailed distribution? Heavy-tailed distribution? Long-tailed distribution? Also, which of the three distributions above is a subclass of the other?
31
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1answer
474 views

Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

I've known that one can arrange all the numbers from $1$ to $\color{red}{15}$ in a row such that the sum of every two adjacent numbers is a perfect square. $$8,1,15,10,6,3,13,12,4,5,11,14,2,7,9$$ ...
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2answers
52 views

Differentiate $f(x) = exp_{a}(x) $ from first principles

Differentiate $f(x) = exp_{a}(x) $ from first principles, for $ a > 0 $ (Recall that $ exp_{a}(x) = exp(x.ln(a)) $ Here is where I am so far: $ f'(x) = \lim\limits_{h \rightarrow 0} ...
0
votes
1answer
48 views

Solving for $z^2 = x^2 -xy + y^2$

Recently, I came across the following solution to finding integer solutions for $z^2 = x^2 - xy + y^2$: $x = k(-n^2 -2mn)$ $y = k(m^2 - n^2)$ $z = k(mn + m^2 + n^2)$ I've been scratching my head ...
2
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2answers
47 views

Modular Arithmetic with Multiple Exponents [duplicate]

I understand how to do modular arithmetic on numbers with large exponents (like $8^{202}$). However, I am having trouble understanding how to calculate something like: $ 3^{3^{3^{3^3}}}$ mod 5 ...
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5answers
45 views

How to use the chain rule to differentiate

I have two examples of problems that I don't know how to differentiate. y = $e^{x^2/3x+2}$ and y = $-10x^{3x^2-4}$ I know to take the ln on both sides. I just don't understand whereto go ...
0
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3answers
30 views

A question in powers

I am a student in the second secondary so I had a question in math about powers like 4 power what gives you half in mathematical way I mean if there is a rule or a theory which can help me if I fall ...
0
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1answer
33 views

Modular Congruence, Power

$(3x^2+2x+1)(4x^3+x^2+5x+1)=5x^5+4x^4+1\equiv 1 \pmod {x^4}$ Expanding the first part, I get $12x^5+11x^4+21x^3+14x^2+7x+1$. However, I do not understand how to get from the above statement to ...
0
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0answers
53 views

Assuming $m \ge 3$, under what conditions is $z^{m-1} = \frac{x^m + y^m}{x + y}$?

One case where this is true is $z=1$: $$(1)^{m-1} = \frac{1^m + 1^m}{1+1}$$ Another case is where $x=y$: $$(x)^{m-1} = \frac{x^m + x^m}{x+x}$$ Assuming that $x,y,z,m$ are positive integers with $z ...
1
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0answers
66 views

Reasoning about $z^n = x^m + y^m$

Let $z,n,x,y,m$ be positive integers with $z \ge 5$ and $m \ge 3$ and $m$ odd. Does it follow that: $z$ cannot be prime if $p \ge 5$ and $p | z$, then either $p > m$ or $p|m$ Here is my ...
1
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1answer
120 views

Thinking about $p^n = x^m + y^m$ where $p | m$

Let $p,m,n,x,y$ be positive integers. Let $p$ be prime and $m$ be odd. if $p \ge 5$ and $m \ge 3$ and $p | m$, are there any solution for: $p^n = x^m + y^m$ I'm not very clear how to proceed on ...
0
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2answers
61 views

Calculate $e^{xA} $

$$ A = \begin{bmatrix} 1 & 1 & -1 \\ -1 & 1 & 1 \\ 1 & -1 & 1 \end{bmatrix} $$ I have the answer, but I don't know the ...
0
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2answers
55 views

When is square root the inverse of the square?

For years I have been wondering about something, and this is the day when the problem shall be rectified forever, by the help of you, ofcourse! :) It seems no one I ask really know the answer. Today ...
0
votes
1answer
51 views

Why reduce $64\bmod{11}$ down to $12\bmod{11}$?

This is the problem I am currently working on Find this value: $(7^3\bmod{23})^2\bmod{11}$ Here's my work: $$\begin{align*} &(7^3\bmod{23})^2\bmod{11}\\ &64\bmod{11}=9 \end{align*}$$ This ...
0
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0answers
33 views

Coordinate Systems on the Complex Plane: Rectangular, Polar, Exponential, … Imaginary?

In the complex plane there is a nice relationship between rectangular, polar, and exponential coordinates: $$(x+iy) = r(\cos\theta + i~\sin\theta) = re^{i\theta} $$ $$where~~x ,y ,\theta, r \in ...
2
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1answer
98 views

Finding remainder of dividing and moding large exponents [duplicate]

Can anyone explain how to calculate the remainder for types of problems like this: $2^{2131312213123}$ divided by 100 $13^{6601}$ mod 77
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0answers
64 views

Can this approach to showing no positive integer solutions to $p^n = x^3 + y^3$ be generalized?

The following problem is a $2000$ Hungarian Olympiad question. Find all primes $p$ such that: $$p^n = x^3 + y^3$$ The answer is that there are only $2$ solutions: $2^1 = 1^3 + 1^3$ $3^2 = 2^3 + ...
3
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0answers
46 views

Is there a notation for exponentiation analog to capital-sigma notation Σ for addition and capital pi Π notation for multiplications?

There are the following common notations: Sums: $$\sum_{i=2}^4 i = 2 + 3 + 4 = 9$$ Products: $$\prod_{i=2}^4 i = 2\times 3\times 4 = 24$$ Is there a (theoretical) one for: Exponentiation ...
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2answers
29 views

Powerseries of $b^x$

I read on wikipedia that the exponential function $e^x$ can be written(defined) in this form $\displaystyle e^x:=\sum_{n=0}^{\infty}({1\over n!})x^n$ So my question was if its is then possible to ...
0
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2answers
32 views

If $m$ is odd, does $\frac{x^m + y^m}{x+y}$ have an odd number of terms?

Let $m,x,y$ be integers with $m$ positive and odd. It seems to me that $\dfrac{x^m + y^m}{x+y}$ will always have an odd number of terms. Here's my reasoning: $\dfrac{x^m + y^m}{x+y} = ...
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3answers
32 views

For odd $m\ge3$, does it follow: $\frac{x^m + y^m}{x+y} + (xy)\frac{x^{m-2} + y^{m-2}}{x+y} = x^{m-1} + y^{m-1}$

Unless I am making a mistake, I am calculating that: $$\frac{x^m + y^m}{x+y} + (xy)\frac{x^{m-2} + y^{m-2}}{x+y} = x^{m-1} + y^{m-1}$$ Here's my reasoning: $\dfrac{x^m + y^m}{x+y} = x^{m-1} - ...
0
votes
0answers
31 views

Sliding window method to calculate power

Actually this is from a homework. I am trying to figure out what is a sliding window method used for calculating power. In python, there is a build in function called ...
1
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3answers
53 views

If $n$ is an odd integer, does $x+y$ divide $x^n + y^n$?

I believe that the answer is yes. Here's my thinking: $x^n + y^n -(x+y)(x^{n-1} + y^{n-1}) = -x^{n-1}y -xy^{n-1}$ $-x^{n-1}y -xy^{n-1} - (x+y)(-x^{n-2}y -xy^{n-2}) = x^{n-2}y^2 + x^2y^{n-2}$ So, at ...
7
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1answer
127 views

Is The Statement $b^n\equiv 1\pmod n$ equivalent to “$x\mapsto b^x-x\pmod n$ is a bijection”?

Suppose that $n$ is a natural number and $b$ is one coprime to it such that $b^n\equiv 1\pmod n$. Does it follow that, if $b^x-x\equiv b^y-y\pmod n$, then $x\equiv y\pmod n$? This is inspired by ...
0
votes
4answers
59 views

Why $2(0.3)^2$ doesn't equal $0.6^2$

Why $2(0.3)^2$ doesn't equal $0.6^2$? I mean if $0.6 = 2(0.3)$, then why $2(0.3)^2$ doesn't equal $0.6^2$? I think it is because of the power but I'm not sure about that. All that I know is that it ...
30
votes
4answers
592 views

What is the *middle* digit of $3^{100000}$?

The decimal representation of $3^{100000}$ has $47713$ digits. What is the $23857^{th}$ digit - i.e. the one in the $10^{23856}$'s place? There are lots of questions on this site asking for the ...
3
votes
6answers
272 views

Finding the mod of a difference of large powers

I am trying to find if $$4^{1536} - 9^{4824}$$ is divisible by 35. I tried to show that it is not by finding that neither power is divisible by 35 but that doesn't entirely help me. I just know that ...
8
votes
6answers
175 views

Why is $\ln(x^x)=x\ln(x)$ valid?

I know that $\ln(x^k)=k\ln(x)$ for any constant $k$, but why is $\ln(x^x)=x\ln(x)$. The exponent $x$ is not constant.
3
votes
6answers
97 views

Is it correct to move x down in $2^x - 2^3 < 0$?

I have $2^x - 2^3 < 0$ and I think it's correct to conclude that $x - 3 < 0$ but a friend of mind disagree with me. I was wondering if there is such a property or axiom?