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

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37 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 ...
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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} = \left(x^{m-1}...
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3answers
33 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} - x^{m-...
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3answers
85 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 ...
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1answer
171 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 the ...
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4answers
70 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 ...
32
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4answers
872 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 last ...
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6answers
344 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 ...
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6answers
208 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.
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6answers
112 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?
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1answer
53 views

Prove that $ exp_{a}(\frac{p}{q}) = \sqrt[q]{a^{p}} \space \forall \space p,q \in \mathbb{Z} $

Prove that $ exp_{a}(\frac{p}{q}) = \sqrt[q]{a^{p}} \space \forall \space p,q \in \mathbb{Z} $ with $ q \geq 2 $ I'm not sure how to approach this question. I was thinking through in induction with ...
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3answers
93 views

Modular Exponentiation 8^5^4 [duplicate]

I am trying to find the last digit of $8^{5^4}$ (or $8^{(5^4)}$, if you will) using modular exponentiation. What I know is that the value I want is: $8^{5^4} \mod 10$. Normally I would find a ...
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3answers
80 views

Modular Exponentiation 3^5^7

I am trying to find the last digit of $3^{5^7}$ (or $3^{(5^7)}$, if you will) using modular exponentiation. Here's what I've figured out: The value I want is $3^{5^7} \mod 10$. $5^n \mod 10 = 5$ (if ...
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2answers
32 views

Integrate a function using variable changement

I did the integraton using Wolfram alpha but I wonder if is it feasible using either integration by part or by substitution
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4answers
64 views

If $a^x=b$, then $ x=$?

Stupid question, I know, but I couldn't remember nor find information by googling on how to find the exponent of $a$ that gives $b$ as the result. If $a^x=b$, then $x=log_a b$ but how do you find $x$?
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1answer
65 views

Deep Roots…:)

Why is it that powers with very small fractional or decimal exponents all tend to one? That is, for $x \ll 1$, $a^x \approx 1$, seemingly. True, or untrue? Can anyone offer more explanation? Thanks ...
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3answers
52 views

Power and exponential equation [duplicate]

So lately I came across this seemingly simple problem that I just can't get around. Solve this equation: $x^2 = 2^x$ I cannot do this algebraically, while I refuse to believe it is impossible to ...
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0answers
57 views

Why continuous growth (based on e) is being simply scaled to match non-limit cases (limit of the (1+1/n)**n formula)?

The constant $e$ is the maximum exponential growth that is possible when it is done continuously, i.e. when what can be called "continuous breeding" occurs. That is the meaning of the $lim_{n\...
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1answer
37 views

how would one solve the following equation

How could the following equation be solved? $$ 100x^2=2^x $$ This is as far as I have got: $$ \ln(100x^2) = \ln(2^x) $$
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1answer
35 views

order of operations with many level exponents

I was wondering, what is the order of operations when it comes to multi level exponents. Couldn't find anything in google. Something like: In this case, if n equals 4, would it be correct to ...
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1answer
67 views

Last digits of n^(n-1)^(n-2)^(n-3) and so on.

If given $n$, how would I get the last digits of $n^{{n-1}^{{n-2}^{\dots}}}$, for example $$5^{\displaystyle4^{3^2}}.$$ As far as I've gotten is that the last digits tend to repeat after a while, but ...
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2answers
54 views
1
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2answers
186 views

Divisibility Proof with Induction - Stuck on Induction Step

I'm working on a problem that's given me the run around for about a weekend. The statement: For all $m$ greater than or equal to $2$ and for all $n$ greater than or equal to $0$, $m - 1$ divides $m^n ...
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1answer
34 views

Evaluating Exponents

Answer choices: 22 15 10 358 None of the above This is my solution:
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3answers
73 views

What $n^{\frac{1}{\log_2n}}$ means?

I was confused with about the $n^{\frac{1}{\log_2n}}$ expression. I am not sure how to make mathematical sense of it - i.e. express it in another way for easier understanding. I tried to plug in some ...
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2answers
34 views

Prove that I can always write a number a, a>0 as any number c, c>0 to the power of some number (a=c^x)

I'm very new to math, I'm sorry if my question is stupid. I started to study math by my own so I can study Computer Engeneering. I'm studying logarithms and I try to come up with simple proofs of the ...
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1answer
57 views

How to simplify logs and powers?

Is there any way to simplify $(\log a)^{\log b} = c$? And even this $(\log x)^y = z$? And also this $(\log m)(\log n) = p$ (which is essentially $\log m^{\log n} = p$) I was trying to simplify some ...
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1answer
117 views

zero raised to power zero in Church encoding

In Church encoding of the natural numbers in lambda calculus raising zero to the power zero gives the answer zero. Does anybody know of an encoding where the answer is 1?
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2answers
4k views

How do I reverse engineer this “power of”/exponent?

Take the following: (2)^3 = 8 I understand that this is 2 * 2 * 2 = 8 My question is how do I reverse engineer this if I ...
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2answers
52 views

Show if $0 \le a <b$ implies $0 \le a^{\frac{1}{n}}<b^{\frac{1}{n}}$

Given that $0\le a<b$ show that $0\leq a^{1/n}<b^{1/n}$ Is this proof by induction? Show it's correct for $n=1$ Assume true for $n=k$, then $0\leq a^{1/k}<b^{1/k}$ holds for some $k$, ...
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2answers
35 views

Why $(10^ab+c)^{4d+1}-c \mid 10$?

I came across the following equation: $$x=(10^ab+c)^{4d+1}-c$$ Why is $x$ a multiple of $10$ for any natural number values for $a$, $b$, $c$ and $d$? The only progress I made was that $a$ could be ...
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0answers
25 views

Exponentiation on order types

How is exponentiation defined on order types? We know that $2^\omega=\omega$. What is $2^{\omega^*}$? Is it $\omega^*$? $\eta$? $\lambda$? I'm guessing $\eta$, but I'm not sure. $\omega$ is the ...
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2answers
25 views

Using the distributive property to factor $(5^{-1}\cdot 5^x - 5^x - 5\cdot 5^x + 5^2\cdot 5^x)$

I can't seem to understand the distributive property. Take this: $$ 5^{-1}\cdot 5^x - 5^x - 5\cdot 5^x + 5^2\cdot 5^x$$ becoming this: $$ 5^x\left(\frac 15 - 1 - 5 +25\right) $$ Help? :D
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3answers
40 views

Beginner exponent/simplification question

Hey there I am having some trouble remembering all the old exponent rules and such, for example, $$ \frac{1}{(6+7^n) ^3} $$ How can I simplify this? I know that (7^n)^3 is the same as (7^3n), but ...
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2answers
46 views

Modular exponentiation

How do you solve: $$5^{{9}{^{13}}^{17}} \equiv x\pmod {11}$$ I've been trying with this but no luck. I get to ${{9}{^{13}}^{17}} \equiv x\pmod {11}$ from $5^3 * 5^3 * 5^3 = 64 \equiv 9\pmod {11}$. ...
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2answers
83 views

Simplifying the exponential expression $e^{-4\ln x +8\ln y +2}$ [closed]

I'm totally stuck on this. Tried numerous sites for a decent explanation but can't find anything. Simplify the expression $$e^{-4\ln x +8\ln y +2}.$$ Thanks in advance.
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1answer
65 views

Quick methods to check perfect$ 4^{th}$,$ 5^{th}$, $6^{th}$ powers

Are there any quick modulus methods to check if a number could be a perfect power (4, 5, 6)? Preferably binary methods. For example, a perfect fourth power has to be $0, 1 \pmod{16}$ from a square ...
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1answer
46 views

What's the solution to this exponential system of equation?

What are the steps to solving a system of equations when $x$ and $y$ are exponents? But they have different base. Here is the problem. $5^x\times3^y=45$ $3^x\times5^y=75$
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1answer
40 views

Help solving simultaneous equation with powers

I am trying to solve the following equations: $0.5 = exp(-(3*c)^k)$ and $0.99 = exp(-(29*c)^k)$ I have used MATLAB to get the answers of $c = 0.21487$ and $k = 0.83471$ but I'd really like to know ...
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3answers
49 views

Are these exponential forms equal?

Is $(\frac1{\sqrt x})^{11}$? the same thing as $x^{\sqrt{11}}$ ? Basically what I'm asking is are those equivalent/the same?
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1answer
52 views

Basic exercise with exponents and radicals

I'm trying to solve a simple high school algebra problem, I would like to know if my result is correct. Convert the radicals into exponents, solve and then express the result as a radical $2\sqrt[3]...
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2answers
37 views

Why do I get two times the base if it's squared when I multiply the value by four?

For example, if I multiply the value of a base squared by four, I also get twice the base if it's squared. Look:$$6^2\cdot4=12^2$$ because $$36\cdot4=144$$and $36$ is the square of $6$ and $144$ is ...
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3answers
339 views

On the sum of digits of $n^k$

Reading another question on the sum of the digits of $2^n$ i started wondering whether there exist a $\alpha\in\mathbb{N}$ such that for every $n>\alpha$ we have $S(2^{n+1})>S(2^n)$, where $S(n)$...
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3answers
250 views

Modulus calculation for big numbers

I am having problems with calculating $$x \mod m$$ with $$x = 2^{\displaystyle2^{100,000,000}},\qquad m = 1,500,000,000$$ I already found posts like this one https://stackoverflow.com/questions/...
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3answers
232 views

How to calculate the sum of digits of $2^n$?

How do I find the sum of digits of $2^n$ in general? Sum of digits of $2^1=2$ is $2$. Sum of digits of $2^{10}=1024$ is $7$. I have check there is no obvious pattern or any recurrence that i can ...
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1answer
86 views

How to prove that $2^x,3^x,5^x\in\mathbb N$ implies $x\in\mathbb N$? [duplicate]

Let $x\in\mathbb R$ and suppose that $2^x,3^x$ and $5^x$ are all integers. Does it imply that $x$ is also necessarily an integer? I read somewhere that the answer is "Yes" and a proof is known, but I ...
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1answer
34 views

Find the square root of a term with a variable

I'm reviewing a PSAT score report with my son and trying to account for the College Board's answer. Below are the question and answer. I follow them as far as: $$ \sqrt{8r^2} $$ From that point, I ...
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2answers
147 views

Solutions of $a^x = x$

How can I find a bound for the solutions of the following equation without using the Lambert function? $$a^x = x,$$ where $a \in \mathbb{R}$.
2
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1answer
151 views

Why doesn't $2^2 = -4$?

I was just curious because a number raised to the $\frac 1x$ where $x$ is an integer greater than $1$ has $x$ solutions, why can't a number to the $x$ where $x$ is an integer greater than $1$ also ...
1
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1answer
113 views

Solving $a=\Big(1+\frac{b}{x}\Big)^x$ for $x$

How to solve this equation for $x$? $$a=\Bigg(1+\frac{b}{x}\Bigg)^x$$ It's not a task that I was asked to solve by someone. I just have to solve it because it's a part of my project. If it's ...