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

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1answer
49 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
77 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
51 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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0answers
15 views

How to steepen logarithmic function without reducing constant of deceleration

As you can see I have plotted my points in Geogebra and compared them to the function $ y=log_{10}x $ They clearly don't coincide, how would I go about adjusting the function in order to find the ...
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2answers
16 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
63 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
44 views

Deep Roots…:)

Why is it that powers with very small fractional or decimal exponents all tend to one? That is, for $x << 1$, $a^x \approx 1$, seemingly. True, untrue? Anyone can offer more explanation? Thanks ...
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3answers
43 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
30 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 ...
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1answer
34 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
10 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
46 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
39 views
1
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2answers
125 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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0answers
19 views

Evaluating Exponents

Answer choices: 22 15 10 358 None of the above This is my solution:
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3answers
66 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
22 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
41 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
80 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
62 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
50 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
13 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
22 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
30 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
40 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
63 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
37 views

Quick methods to check perfect 4th, 5th, 6th 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 $0, 1 \pmod 8$ from a square ...
1
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1answer
29 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
35 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
48 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
30 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 ...
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2answers
27 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
264 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 ...
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3answers
89 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 ...
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3answers
132 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 ...
3
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1answer
84 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
31 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
128 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 $a \in \mathbb{R}$.
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1answer
148 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 ...
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1answer
111 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 ...
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0answers
81 views

Algorithm to calculate powers

Is it possible to write an algorithm that uses only multiplication and addition to calculate $a^b$ where both a and b are real numbers?
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1answer
105 views

$x^n y^n = (xy)^n$, proof exercise

As an exercise, I tried to prove the following theorem. Please share your thoughts about what I wrote. (The proof only uses the utensils which are listed below.) Theorem \begin{equation*} x^n ...
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1answer
170 views

Existence of $x$ such that $2^x =a,3^x=b,5^x=c$ for some integers $a,b,c$

Conjecture: There does not exist a non-integer $x$ such that $$2^x=a$$ $$3^x=b$$ $$5^x=c$$ where $a,b,c$ are all integers. I'm aware that the similar question There does not ...
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0answers
108 views

How many zeroes would be there at the end of $11^{(5!)!}-1$?

$$11-1=10 \\ 121-1=120 \\ 1331-1=1330$$ Now it can be seen that the tens digit increases by 1 at each increment of exponent. So, only in case of $11^{10}$ the tens digit is zero and the units digit ...
0
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1answer
48 views

Find exponential function given two points

$f(x) = ar^x $ given that $r > 0$. I'm given two points ($3$, $\frac{8}{9}$) and ($4$, $\frac{16}{27}$). My textbook then says $$r = \frac{\frac{16}{27}}{\frac{8}{9}}$$ Why does this work? I ...
6
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1answer
105 views

How to prove $x^ax^b = x^{a+b}$

I am looking for a proof of one of the exponent combination laws, namely the sum of powers. Here $x, a, b \in \mathbb R$ and $x > 0$. I thought about induction but since a,b are not only positive ...
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2answers
131 views

Exponential congruence

Hi All am a bit stuck on some revision that I am trying to do. Firstly (part a) I must calculate the inverse of 11 modulo 41, which I have done and believe it to be 15. The next part is to: Now use ...
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2answers
143 views

Powers containing every digit equally often

There are several nontrivial powers containing every digit equally often, for example $32043^2$ $2158479^3$ $69636^4$ $643905^5$ $3470187^6$ A necessary condition for a power with the desired ...
3
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3answers
128 views

Proof that sum of first $n$ cubes is always a perfect square [duplicate]

I know that $$1^3+2^3+3^3+\cdots+n^3=\left(\frac{n(n+1)}{2}\right)^2$$ What I would like to know is whether there is a simple proof (that obviously does not use the above info) as to why the sum of ...