Questions tagged [exponentiation]

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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3answers
61 views

How can $x^0=1$ be proved with a real life experiment?

I saw that $x^0=1$ proof is $\displaystyle x^{a-a} = \frac{x^a}{x^a} = 1.$ However I don't find this a convincing proof. How can someone prove this without using the fraction above? or, Is that ...
0
votes
1answer
48 views

I need to make an exponental equation with subtracting

I have this equation $$X(1+0.2)^Y$$ Which is simply adding 20% every $Y$ month for the $X$. For example $$100(1+0.2)^1 = 120$$ $$100(1+0.2)^2 = 144 $$ What I want to do is cutting or rebating 10 ...
0
votes
0answers
34 views

Find the exponent of a number with base 2

I am trying to write a program for which I need to map the following data: in out 1 -> 0 2 -> 1 4 -> 2 8 -> 3 16 -> 4 etc. I've figured out ...
5
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0answers
72 views

Do cyclic groups appear as symmetry groups of exponentiation-only terms?

Let $\mathfrak{E}=(\mathbb{N};\mathsf{exp})$ be the structure consisting of the natural numbers together with exponentiation. Given a term $t$ in the language of $\mathfrak{E}$ - that is, an ...
0
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1answer
91 views

An algebra question from usa math contest [closed]

Question- if k is even number then for what value of x and y , $$x^{2k} - (xy)^k + y^{2k} =0$$ I dont have any knowledge about this math, i generally solve normal math . Please anyone give me a hint ...
0
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0answers
26 views

Powers of Fourier series

I have a problem with the Fourier series of a Jacobi elliptic function. Let us assume $$ \operatorname{sn}(x,i)=\sum_{n=0}^\infty a_n\sin\left((2n+1)\frac{\pi}{2K(i)}x\right) $$ with $a_n$ known (...
3
votes
1answer
116 views

Is $\displaystyle\sum_{n=1}^{\infty} \frac{1}{2^n+1}$a complex number? What is happening?

While computing the integral $$\displaystyle\int_0^1{\displaystyle\sum_{n=1}^{\infty}x^{(2^n)}dx}$$ I easily got to $$\displaystyle\sum_{n=1}^{\infty} \frac{1}{2^n+1}$$ Since this was getting ...
-2
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1answer
79 views

Why is $3^{\sqrt {3}}$ and $27^{1/2}$ not equal? [closed]

Maybe I am missing something really basic here. But surely, $3^{\sqrt {3}}$ is $3^{3^{1/2}}$. And that is equal to $27^{1/2}$. However, evaluating $3^{\sqrt {3}}$ doesn’t give the same answer as $27^{...
0
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0answers
25 views

Exponential difficulty

I am finding it difficult to do the $(2(2^{-x})$ part of the problem below. So far I have gotten $2^{x+1}$ to equal $2y$ and I cannot seem to find how to get $2(2^{-x})$ $2^{x+1} + 2(2^{-x}) -5 = 0$ $...
0
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0answers
45 views

Solving equation where unknown is on the exponent

In general is there a systematic of solving equations of the following form: $$ a \exp(bx^2+cx)+ d \exp(x) = e $$ where $a$, $b$, $c$, $d$, $e$ are constants and $x$ is the unknown.
-2
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1answer
79 views

Is it correct that $b^a<a^b$, where $b>a>e$? And how do I prove it? [closed]

After watching many videos about "which number is bigger?", I decided to make the following inequality: $$b^a<a^b$$ where $b>a>e$. Is this inequality correct? If it's not correct, ...
0
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1answer
17 views

Prove that, if for all eigenvalues $\lambda$ of $A$ $Re \: \lambda < -\mu$, we have $||Ax||_{op} \leq C_0 e^{-\mu x} \; \forall x > 0$

Say $A \in L(\mathbb{R}^n, \mathbb{R}^n)$, and for all eigenvalues of $A$ (let's call them $\lambda$) we have $Re \: \lambda < -\mu$. I need to prove that $||Ax||_{op} \leq C_0 e^{-\mu x} \; \: \...
0
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1answer
42 views

Simplify $A=\frac{y^\frac12}{y^\frac12-2}+\frac{y^\frac12}{y^\frac12+2}-2$

Simplify $$A=\dfrac{y^\frac12}{y^\frac12-2}+\dfrac{y^\frac12}{y^\frac12+2}-2$$ So $$A=\dfrac{y^\frac12\left(y^\frac12+2\right)+y^\frac12\left(y^\frac12-2\right)}{y-4}-2=\dfrac{2y^\frac14}{y-4}-2=\...
0
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1answer
11 views

Cancel out $\dfrac{a\left(a-b^{\frac12}\right)}{a+a^{\frac12}\cdot b^{\frac14}}$ [duplicate]

Cancel out $$\dfrac{2-54b}{2-6b^\frac13}$$ I really don't see what we are supposed to do. This is what I have tried to do with the numerator $$2-54b=2-9\cdot6b=2-3^3\cdot2b=2(1-3^3b)$$
0
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1answer
26 views

Cancel out $\frac{a\left(a-b^{\frac12}\right)}{a+a^{\frac12}\cdot b^{\frac14}}$

Cancel out $$\dfrac{a\left(a-b^{\frac12}\right)}{a+a^{\frac12}\cdot b^{\frac14}}$$ The given expression is equal to $$\dfrac{a\left(a-b^{\frac12}\right)}{\left(a^\frac12\right)^2+a^\frac12\cdot b^\...
0
votes
2answers
63 views

Under which conditions $-(-c)^{-n}c^n=(-1)^{1-n}$? [closed]

Set $c>0$ and define $0\leq n \leq m$ and integer. Do we need certain conditions to be satisfied so we have $$-(-c)^{-n}c^n=(-1)^{1-n}?$$ When we try to simplify the LHS in Mathematica, we don't ...
-1
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3answers
35 views

Is $a^x \pmod{n} = (a\pmod{n})^{x \pmod{n}}$? [closed]

Is it right to claim that: $a^x \pmod{n} = (a \pmod{n})^{x \pmod{n}}$? I know that: $a^x \pmod{n} = (a \pmod{n})^{x}$?
0
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0answers
45 views

Matrix Exponentiation in Olympiad Problem [duplicate]

I've seen a problem that gives $$A= \begin{bmatrix} 4 & -\sqrt5\\ 2\sqrt5 & -3 \end{bmatrix}$$ and asks to find all pairs $(n,m) \in \mathbb{N} \times \mathbb{Z}$, with $|m|\leq n$ such that: ...
3
votes
1answer
74 views

What is the highest number of digits so that this number of digits in a specific power of 2 are exactly 10%?

I accidentally found out that in $2^{{10}^{6}}$ (==$2^{1000000}$), there are exactly 10% digits of 6 (in the decimal form). And I would like to know - are there powers of 2 in which all the digits ...
0
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3answers
49 views

Simplifying $x \gt x^{\ln 2/\ln 3}$ algebraically to $x \gt 1$

I simplified an inequality to the following point $x > x^{\ln 2/\ln 3}$, I know the answer is $x > 1$. Of course one could explain that $y = x$ exceeds $y = x^n$ where $0<n<1$ at $x>1$ ...
0
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0answers
28 views

Binet's Formula for Negative Fibonacci Sequences

I have recently learnt the Binet's formula for calculating Fibonacci Sequences and got my mind blown. $F_n = \frac{(1+\sqrt{5})^n-(1-\sqrt{5})^n}{2^n \sqrt{5}}$ This has worked charm for positive ...
0
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1answer
17 views

Resolve to n in net present value formula

There is the formula for the net present value which consists of a few variables and also of exponents. Resolving to $K$, $C$, and $A_0$ is fairly simple. But is there a possibility to resolve to / ...
0
votes
2answers
54 views

Matrix operation to exponentiate each element in a vector

I am using the following matrix algebra to obtain a vector, however, I eventually need all the resulting elements to be exponentiated. \begin{equation} \begin{split} \boldsymbol{\beta}^{\textsf{T}}\...
0
votes
2answers
27 views

Multiplication of variables to power

I got this equation $$MRS= - \frac{3/4x^{1/4} \ y^{-1/2}}{\frac{}{}3/2x^{-1/4}\ y^{1/2}}$$ As you can see the absolute values of the exponents are the same so they should cancel out each other somehow,...
2
votes
1answer
27 views

Exponentiating expression containing ln(abs(x))

I am trying to figure out when we write +/- after exponentiating expressions containing natural log. So, say that we have integrated (1/x) with respect to x. Then we have ln(abs(x)) + C. That is, ln(x)...
1
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3answers
47 views

Evalute $\sqrt[n]{\frac{20}{2^{2n+4}+2^{2n+2}}}$

Evaluate the given expression $$\sqrt[n]{\dfrac{20}{2^{2n+4}+2^{2n+2}}}$$ The given answer is $\dfrac{1}{4}$. My attempt: $$\sqrt[n]{\dfrac{20}{2^{2n+4}+2^{2n+2}}}=\sqrt[n]{\dfrac{20}{2^{2n}\cdot2^4+2^...
2
votes
2answers
64 views

About the property $(a^m)^n=a^{mn}$ when $a<0$

I found something I haven't noticed before while solving the following problem: $$\left(9a^2b^\frac12c^\frac13\right)^\frac12\text{ }\color{red}{(b,c\ge0)}$$ I said, okay we can use the property $(...
0
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0answers
48 views

How to prove that $(a^b)^c = a^{(bc)}$ for rational and negative exponents?

I have this question about exponentiation. How can one prove that $(a^b)^c$ = $a^{(bc)}$ for rational and negative exponents? Firstly, we need a basis of what a rational and negative exponent are. So, ...
7
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0answers
147 views

What are the symmetry groups of exponentiation-only terms?

Now asked at MO with slightly different language. Let $\mathcal{E}=(\mathbb{N};\mathit{exp})$ be the algebra (in the sense of universal algebra) consisting of the natural numbers with just ...
0
votes
1answer
43 views

Why is this not the correct way to treat this exponent? [closed]

This is probably a silly question but why is the following not true: $x^\frac{-2}{3}=x^{-1.\tfrac{2}{3}}=x^{-1}x^{\tfrac{2}{3}}=\tfrac{1}{x}.x^{\tfrac{2}{3}}=\tfrac{(x^2)^{1/3}}{x}$
0
votes
1answer
90 views

$k^t=8$, $t^k=9$

I have $k^t=8$ and $t^k=9$. The solutions are $2^3=8$ and $3^2=9$ but I Can't think of a way to solve it without knowing the answers. I've tried $t=\frac{ln(8)}{ln(k)}$ and then $(\frac{ln(8)}{ln(k)})^...
1
vote
1answer
60 views

Power of Stochastic Matrix

Given the stochastic matrix $$Q = \begin{pmatrix}0&2/3&1/3\\1/3&0&2/3\\2/3&1/3&0\\\end{pmatrix}\in \mathbb{R}^{3\times3}$$ I wish to compute $Q_{1,1}^n$ (the entry in the first ...
1
vote
4answers
66 views

Vanier College Practice test - Prove that $\log_{\frac {1} {\sqrt b} }\sqrt x= -\log_b x$

Source : Colorado.edu, Vanier College, Worksheet Logarithm Function, Question 8 (https://math.colorado.edu/math1300/resources/Exercises_LogarithmicFunction.pdf) To be proved : $\log_{\frac {1} {\sqrt ...
0
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0answers
26 views

How do negative powers and fractional powers make sense? [duplicate]

We know that powers are the number of times a number is multiplied with itself. Like $2^3$ means $(2×2×2)$, $5^4$ means $(5×5×5×5)$, etc. But how do negative powers make sense? What does it mean to ...
3
votes
1answer
104 views

How is $ai^{bi}$ a real number?

$ai^{bi}$ 1 immediately appears as an imaginary number: How could an imaginary to the power of yet another imaginary not result in an imaginary? But, for example, $3i^{2i}≈0.12964$ (truncated). How is ...
3
votes
1answer
99 views

Exponent law for complex numbers

Wikipedia states for $z\in\mathbb{C}$ and $n\in\mathbb{N}$ ${\displaystyle (z^{n})^{1/n}\neq z}$ Is this correct? In general, the exponent law $(x^a)^{1/b}=x^{a/b}$, $a,b\in\mathbb{N}$ for nonnegative ...
0
votes
1answer
100 views

How can I retrieve $a$ and $b$ from $2^a \cdot 3^b\,$?

When given a number $k$ in the form $2^a \cdot 3^b$ with integers $a,b$ , I want to use a function $f(k)$ to retrieve $a$ and a function $g(k)$ to retrieve $b$. I know that prime factorization can ...
1
vote
0answers
32 views

Wolfram Alpha computation of exponents

I am a bit confused by a Wolfram Alpha. In my calculations I had a term like $x^{(3/2)}$ which I wrote as $\sqrt{x^3}$. As silly mistakes happen quite often to me, I often double check stuff like this ...
-3
votes
1answer
43 views

Trouble identifying error in fake-proof. [duplicate]

I was talking to a friend about the problem with this proof and I'm stomped on the illegal step: $i = (-1)^{(1/2)}= (-1)^{(2/4)} =((-1)^2)^{(1/4)} = 1^{(1/4)} = 1$
-1
votes
2answers
49 views

Finding the value of x in equations involving exponents [closed]

The equation is $(6x)^{3x} = 6^{36}$ Find the integer value of x. By hit-and-trial method, we can find that x = 6 But, how to solve the equation mathematically.
0
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0answers
27 views

How to evaluate rational numbers to rational powers

Is there a method besides a Taylor series to calculate $x^y$, where $x$ and $y$ are rational numbers? So for example, I require a method to calculate $2.128321\dots^{5.3212\dots}$ to a certain amount ...
0
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0answers
23 views

Why is 1 raised to the power of infinity undefined? [duplicate]

The problem asked to find the following limit $$\lim_{x\rightarrow0}(cos(x))^{\frac{1}{x}}$$ which I found was equal to $1$ using the limit chain rule. However the question made me wonder why $1^\...
2
votes
2answers
94 views

Why am I getting 2 different answers for Integral exp(ix) from 0 to Infinity

Hi I was trying to solve this integral: $\int_{0}^{\infty}e^{ix}dx$ When I integrate it normally and apply limits, I get an undefined answer as $e^{\infty}=\infty$ But if I take $i$ as $-\frac{1}{i}$, ...
1
vote
1answer
59 views

Find all integers $n, n\gt2$ such that $n^{n-2}=x^n$ for some $x$

We can express this alternatively as $n^{n}=n^{2}x^{n}$. So the number raised to the power of itself has to be proportional to some number to the $n$-th power, but cannot be equal, naturally. I am not ...
2
votes
3answers
65 views

Is there a minimum set of properties that uniquely defines complex exponentiation?

Starting with real numbers, the four basic operations, and limits, we can define exponentiation using the properties: $a^0 = 1$ (base case for our purposes) $a^1 = a$ (base case) $a^{b+c} = a^ba^c$ (...
3
votes
1answer
87 views

What's the connection between $\exp_\text{Lie}$ and $\exp_\text{Spectral}$?

There are (at least) two places in quantum mechanics where we commonly exponentiate operators. Lie groups: We exponentiate elements of a Lie algebra, yielding elements of the associated Lie group. E....
3
votes
2answers
116 views

A question relating to exponent laws

Here is the question: If: $$125*(3^x) = 27*(5^x)$$ Then find the value of $x$. Here is what I have done so far: I found that I can manipulate the numerical values to get the same bases: $\to 125 = 5^3$...
1
vote
0answers
54 views

Is there any power of $4$ with natural exponent that can be represented as sum of several different powers of $5$ with whole non-negative exponent?

There is a power of $2$ with a natural exponent, which can be represented as several different powers of $3$ with whole non-negative exponent. For example, $2^{2}=3^{1}+3^{0}$ or $2^{8}=3^{5}+3^{2}+3^{...
4
votes
1answer
107 views

Exponentiation in Lambda Calculus

I just need a double check. Some recitation notes say: EXP $= λmn.m$ $($MUL $n)$ $\underline{1}$. The notes also say: EXP $= λmn.nm$ I think I found an inconsistency and would like someone to double ...
4
votes
1answer
177 views

$~A:=\text{matrix} ~\rightarrow~\lim_{n\to\infty}A^{n}=?~;~$How should I approach against it first?

$$A:=\begin{pmatrix} \alpha&1-\alpha\\1-\beta&\beta\\\end{pmatrix}$$ $$\left(0<\alpha,\beta<1\right)~~\wedge~~\left(\alpha+\beta\neq 1\right)$$ $$\underbrace{\lim_{n\to\infty}A^{n}}_{\...

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