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

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1
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2answers
18 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 ...
1
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2answers
47 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 $k$, ...
0
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2answers
32 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 ...
1
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0answers
7 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 ...
1
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2answers
37 views

Limit and Fibonacci [on hold]

How to prove $${\lim_{n \to \infty} \frac{F_{kn}}{{F_n}^k} = 5^{(k-1)/2}}$$ Non-induction method is prefered.
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2answers
21 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
0
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3answers
27 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
37 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 $5^3 * 5^3 * 5^3 = 64 \equiv 9\pmod {11}$. ...
0
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2answers
42 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.
1
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1answer
31 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. $5^x\times3^y=45$ $3^x\times5^y=75$
0
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0answers
36 views

is the definition of fractional exponents correct?

I have seen on the internet the following written as the definition for rational exponents: $$a^{\frac{m}{n}} = \sqrt[n]{(a^{m})} = (\sqrt[n]a)^m$$ where no restriction ever seems to be stated that ...
0
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1answer
30 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 ...
1
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3answers
46 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?
1
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1answer
28 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
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2answers
25 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 ...
11
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3answers
244 views

On the sum of digits of $n^k$

Reading another question on the sum of the digits of $2^n$ i started wondering wether there exist a $\alpha\in\mathbb{N}$ such that for every $n>\alpha$ we have $S(2^{n+1})>S(2^n)$, where $S(n)$ ...
0
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3answers
69 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 ...
2
votes
3answers
117 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
77 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 ...
1
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1answer
29 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 ...
2
votes
2answers
123 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
147 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
110 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 ...
0
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0answers
62 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?
3
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1answer
97 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 ...
4
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1answer
163 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 ...
-1
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0answers
26 views

Is it correct to say “Rational exponentiations are continuous”?

I am not a native speaker and I want to know how to say that this function: \begin{align*} \mathbb{Q} &\to \mathbb{Q} ^*_+\\ x &\mapsto a^x \end{align*} is continuous on Q. I just asked ...
2
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0answers
98 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
37 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
97 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 ...
1
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2answers
113 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 ...
5
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2answers
141 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
101 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 ...
0
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2answers
49 views

What is <any number>^i?

I think I understand what imaginary numbers are, that $i$ is basically the name we give to $\sqrt{-1}$. Does $n^i$ have any sort of meaning? Is it used for anything? You can't really multiply $n$ by ...
25
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2answers
304 views

Generalisation of $(n+3)^2-(n+2)^2-(n+1)^2+n^2=4$

After seeing the neat little identity $(n+3)^2-(n+2)^2-(n+1)^2+n^2=4$ somewhere on MSE, I tried generalising this to higher consecutive powers in the form $\sum_{k=0}^a\epsilon_k(n+k)^p=C$, where $C$ ...
0
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2answers
18 views

A perimeter and exponentiation problem

A wheel of a car lasts for 5000 scandinavian miles, and then it needs to be switched. The diameter of the car's wheel is 70 centimetres. How many laps has the wheel spinned before it needs to get ...
5
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2answers
86 views

Simplify $x ^ y + x ^ z$ to a formula with only one $x$

Is there a way to simplify $x ^ y + x ^ z$ to a formula with only one $x$? I know $(x ^ y)(x ^ z) = (x ^ {y + z})$, but how can it change in addition?
4
votes
7answers
376 views

Integral evaluation (step-by-step)

I'm trying to evaluate the integral by exponent. Could you help me with following steps? Integral: $$\int \frac{1}{4+\sin(x)} dx$$ $$\sin x=\frac{e^{ix}-e^{-ix}}{2i}$$ $$\int \frac{1}{4+sin(x)} dx ...
15
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3answers
270 views

How can I prove $\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}…}}}}=2$ [duplicate]

How can I prove $$\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}...}}}}=2$$ I don't know which method can be used for this?
4
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0answers
88 views

Calculate $1^1 + 2^2 + 3^3 + … + n^n$

Is there a formula to calculate $1^1 + 2^2 + 3^3 + ... + n^n$ I searched but didn't find a formula for increasing powers
3
votes
3answers
65 views

Calculations using googolplexes

How can I calculate $\dfrac{10^{10^{100 }}}{ 10^{10^{70}}}$? I have tried using logs ie: $$\frac{10^{10^{100}}}{10^{10^{70}}}$$ $$=\frac{(100\times \ln(10)) \times \ln(10)}{(70\times \ln(10)) ...
2
votes
1answer
41 views

Find the leftmost (most significant digits) of a large exponent calculation, say $99^{99}$

I want to find the initial 10 digits of an exponent calculation whose result is a very large number - Say, $99^{99} = 3.697296 \times 10^{197}$ I only need to know the digits $3697296$ Is there any ...
7
votes
3answers
144 views

How to visualize $f(x) = (-2)^x$

Background I teach Algebra and second year Algebra to middle school students. We are currently studying Exponential, Power, and Logarithmic functions. We study exponential functions (of the form ...
0
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2answers
31 views

Rules regarding exponents

Given the following algebra problem: $$2^{n+1}-1+2^{n+1}=2^{n+1+1}-1$$ I know $2^{n+1}=2^n2^1$ but just to confirm the truth of the problem above, I just assumed the left hand side is $2^{n+2}-1$ ...
5
votes
3answers
108 views

Rules for whether an $n$ degree polynomial is an $n$ degree power

Given an $n$ degree equation in 2 variables ($n$ is a natural number) $$a_0x^n+a_1x^{n-1}+a_2x^{n-2}+\cdots+a_{n-1}x+a_n=y^n$$ If all values of $a$ are given rational numbers, are there any known ...
2
votes
4answers
123 views

Calculating power without using a calculator, for example $1.05^{10}$

How to find (or estimate) $1.05^{10}$ without using a calculator? Do we have any fast algorithm for cases where base is slightly more than one? Say up to $1.1$ with tick $0.05$.
1
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4answers
40 views

Can we add fractional powers of negative numbers?

This question might be silly and very basic. But my friend and me happened to argue on this for long. My argument was, if $-2 \sqrt3=\sqrt{12}$ which came from $\sqrt{(-2)(-2)} \sqrt{3} $ . If this is ...
-3
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1answer
55 views

How to prove exponent laws for various number systems, including real exponents

How could I prove following exponent laws for set of real, in the given order? 1) $a^m*a^n=a^{m+n}$ CaseI a^m=a.a.a...to m factors a^n=a.a.a...to n factors a^ma^n=a.a.a...to m+n factors ...
0
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2answers
37 views

Converting 29^1312000 to base 10

I am trying to do some calculations with the number 291312000 and I find it would be much easier if I could convert it (approximately) to a base 10 number. The closest I could come was to start with ...