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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3answers
76 views

How to derive that for every real $y > 0$, for every positive real $z \neq 1$, there is a $x \in \mathbb{R}$ such that $y=z^x$.

I am not sure on how to derive the following statement concerning the reals (that I think should be true). For every real $y > 0$, for every positive real $z \neq 1$, there is a $x \in ...
2
votes
3answers
89 views

Why does the minimum value of $x^x$ equal $1/e$?

The graph of $y=x^x$ looks like this: As we can see, the graph has a minimum value at a turning point. According to WolframAlpha, this point is at $x=1/e$. I know that $e$ is the number for ...
0
votes
0answers
21 views

Degree of a root term?

I had the textbook question: What is the degree of the following expression: $x^2\sqrt{y-5}$ Would it be 2.5, since it would be the sum of the exponents of $x$ (2) and $y$ (.5, I think)? Or ...
2
votes
1answer
62 views

Let $0<a<1$, for which $x>0$ is $x^{a^x} = a^{x^a}$ true.

Let $0<a<1$, for which $x>0$ is $x^{a^x} = a^{x^a}$ true. This is where I have gotten so far: $\log_a( x^{a^x} )= \log_a( a^{x^a} ) $ $ a^x \log_ax = x^a $ Now, I know I am only ...
4
votes
1answer
77 views

How is exponentiation defined?

Here is how I think this work: We define $ a^b = \underbrace {a\cdots a}_b $ for $a \in R $ and $b \in N$. Since so far we have not defined what $ a^{-1}$ is, $a^{-4}$ makes no sense. (right?) We ...
1
vote
2answers
37 views

Square of a Sequence of Numbers

This is a simple question, but I could not find the solution. What is the compact form of expansion of $(n_1+n_2+\ldots+n_k)^2$ Is it: $\sum_{i=1}^k n_i^2 + 2\sum_{i=1}^{n-1} ...
1
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2answers
53 views

Prove $e^{i\alpha\hat{X}}=\cos(\alpha)I+i\sin(\alpha)\hat{X}$ if $\hat{X}^{2}=I$.

If we have an operator $\hat{X}$ such that $\hat{X}^{2}=I$ (the identity), how do we prove that: $$e^{i\alpha\hat{X}}=\cos(\alpha)I+i\sin(\alpha)\hat{X} \ ?$$
1
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0answers
25 views

Problem with custom made natural log and power functions

I have made these two functions with the help of posts on math.stackexchange.com. For ln I'm using information gathered from Calculate Logarithms by Hand and for ...
-6
votes
2answers
83 views

How to simplify surds [closed]

Let $\sqrt\frac{\sqrt\frac{\sqrt\frac{\sqrt\frac{\sqrt\frac{\sqrt2}{3}}{4}}{6}}{8}}{9}$ $=$ $\frac{2^m}{3^n}$ Find the value of $mn$.
0
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5answers
95 views

Can't solve this exponential equation: $5^{x+1}-3\cdot 5^{x-1} - 6\cdot 5^x+10 = 0$ [closed]

How does one solve for $x$ in the following: $$5^{x+1}-3\cdot 5^{x-1} - 6\cdot 5^x+10 = 0$$
0
votes
1answer
42 views

How to handle indices with fractional degree?

An algebra problem ate my head!!! $x$ and $y$ are positive real numbers such that $$\sqrt{x^2 + \sqrt[3]{x^4 y^2}} + \sqrt{y^2 + \sqrt[3]{x^2 y^4}} = 512.$$ Find $x^{2/3} + y^{2/3}$. It ...
-1
votes
2answers
66 views

How many times is $3^{20} + 3^{22}$ greater than $3^{20}$ [closed]

I just don't know where to start any help is appreciated.
2
votes
1answer
48 views

How to find $a^b$, where $a$ and $b$have more than $10$ digits?

Consider any two numbers $a$ and $b$ of more than 10 digits, how to find $a^b$ (without the aid of computing devices). Is there any shortcut method to do it. other than binomial series. How do I solve ...
1
vote
3answers
75 views

Is there a way to calculate decimal powers using only addition, subtraction, multiplication and division?

I am a programmer who is trying to build an arbitrary precision decimal class library for C# and was successful in implementing all the basic operations like addition, subtraction, multiplication and ...
0
votes
2answers
32 views

is tg^-1 (x) not the same as tg(x)^-1?

Well, for better syntax, my question looks like this: Isn't $tg^{-1}(x)$ the same as $(tg(x))^{-1}$ ? I always thought that it was absolutely the same. But I was trying to solve a task on an ...
2
votes
1answer
24 views

Proof for the behavior of both types of improper integrals for different powers of x

I was trying to prove for what values of p eq.1 converges or diverges, they didn't give the proof for eq.1 but for eq.2 a proof was given and when I was done with the proof for eq.1 I noticed that for ...
1
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1answer
41 views

Evaluation of $x^{y^{z}}$

Whether $x^{y^{z}}$ should be considered as $x^{\left ( y^{z} \right )}$ or $\left ( x^{y} \right )^{z}$, without any context? If any one among these two is default consideration? $\left ( x^{y} ...
1
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3answers
60 views

Find base of exponentiation

Given the two primes $23$ and $11$, find all integers $\alpha$ such that $\alpha^{11} \equiv 1 \mod 23$. How to compute this? What to use?
4
votes
3answers
110 views

Showing different definitions of exponentiation are equivalent

Suppose we define $\exp(x)$ as the unique function $f:\mathbb{R} \rightarrow \mathbb{R_+}$ satisfying $f(0) = 1$ and $f'(x) = f(x)$ for all $x \in \mathbb{R}$. We then define its inverse ...
1
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4answers
67 views

Raising to rational power - issues

Raising a real number to a rational power is very simple, right? Consider the following example: $$−27 = (−27)^{\frac{2}{3}\frac{3}{2}} = ((−27)^{\frac{2}{3}})^\frac{3}{2} = 9^\frac{3}{2} = 27$$ The ...
2
votes
2answers
133 views

Could you solve $x↑^n2=x↑^m2$?

As my title asks, could you solve $x^2=2^x$? But that's the worrisome part, as I noticed $x↑^n 2=x↑^m 2$ and $2↑^p x=2↑^q x$ will always have a solution at $x=2$. However, there is bound to be at ...
1
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0answers
14 views

How to simplify a sum for the total cost of a yearly payment including compound interest

I want to simplify the below sum for the total cost over a yearly payment including compound interest over n years. An example: we have 150 euros that need to be paid every year and an interest of ...
0
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2answers
21 views

Clarification regarding multiple modular exponentiation

If the base is same and exponents are different, for e.g. R1=b^x mod p; R2=b^y mod p; R3=b^z mod p; (p is large prime (2048 bit); x, y and z - 160 bit integers)) To calculate R1, R2 and R3 at the same ...
4
votes
1answer
121 views

Inverse of $f(x)=3^x+2^x$

I'm tring to find inverse of $f(x)=3^x+2^x$ but I don't have any clue. I tried to $$y=2^x((3/2)^x+1)$$ $$\ln y=\ln2^x+\ln((3/2)^x+1)$$ $$\ln y= x \ln2+\ln((3/2)^x+1)$$ but I can't continue
0
votes
1answer
27 views

Modular exponentiation commutativity in Diffie-Hellman

I've been learning about Diffie-Hellman key exchange. One of the main tricks comes down to a commutativity property of exponentiation in the relevant modular arithmetic, it seems. Something like: ...
7
votes
1answer
122 views

Is $a^{\ln b} = b^{\ln a}$?

I was struggling with a math problem, namely, a limit with a power to the log of something. While I was struggling with it, I found out that $$a^{\ln b} = b^{\ln a}$$ for all positive values that I've ...
2
votes
1answer
22 views

Is there a term in mathematics for Metcalfe's Law?

Metcalfe's Law states that the value of a network is proportionate to the square of the number of users. This comes from the idea that there are $N*(N-1)/2$ pairs in a network of size $N$. Does this ...
2
votes
1answer
23 views

General formula for exponent of a semidirect product

This page states that Exponent of semidirect product may be strictly greater than lcm of exponents But it doesn't give any proof for that. Could anyone provide a general formula for the ...
0
votes
1answer
39 views

Solve $g(x + \frac{1}{x}) = x^y + \frac{1}{x^y}$

Let $y$ be a real number. Find $g$ such that $$g(x + \frac{1}{x}) = x^y + \frac{1}{x^y}$$ Is valid for all real $x$.
-2
votes
1answer
42 views

Simplifying Exponents (exponent laws) [closed]

So I have this equation: $${\left(\left(\frac{w}{x}\right)^{\frac{y}{x}}\right)}^{\frac{z}{x}}$$ And I know that $${\left(a^b\right)}^c = a^{bc}$$ So I figured I'd simplify it into this: ...
2
votes
1answer
76 views

If $q, r \in \mathbb{R}, x \in \mathbb{R}^+$ then $(x^q)^r=x^{qr}$

I'm stuck on this exercise from Tao's Analysis 1 textbook: show that if $q, r \in \mathbb{R}, x \in \mathbb{R}^+$ then $(x^q)^r=x^{qr}$. DEF. (Exponentiation to a real exponent): Let $x>0$ be ...
2
votes
0answers
39 views

Integral of the product of a power function and an arbitrary exponentiated function

I was only able to find integral tables that solve $$f(t)=\int t^c e^{kt}dt$$ but the integral I'm trying to solve has a function, not a constant, for the exponent: $$f(t)=\int t^c e^{g(t)}dt$$ Is ...
3
votes
3answers
77 views

Is $(-1)^{ab} = (-1)^{ba}$ true? => $(-1)^{ab} = ((-1)^a )^b$ is true? [duplicate]

In general we know $A^{bc} = A^{cb}$ for integer $A$. I want to extend this to the case $A=-1$. For integers $a,b$ I guess the above relation holds, \begin{align} (-1)^{2\cdot3} = ((-1)^2)^3 = 1 = ...
1
vote
1answer
38 views

General formula to compute the exponent of the symmetric group $S_n$

Someone has already asked whether an exponent less than $n!$ is possible for a symmetric group $S_n$. It has been answered that it is for $n \ge 4$. I would like to know if there is a general ...
5
votes
4answers
71 views

Solve the equation: $2^{2x+1}=\left(\frac{1}{32}\right)^x$

Having trouble with this problem: $$2^{2x+1}=\frac{1}{32^x}$$ Do I need to set the exponents equal to each other?
-2
votes
1answer
119 views

Please help me solve real-analysis problem [closed]

Problem: Assume we have the next recursive sequence: $$\begin{cases}x_n=\sqrt[3]{6+x_{n-1}}\\x_1 = \sqrt[3]{6}\end{cases}$$ Prove that there exists a constant $C \neq 0$ such that: ...
3
votes
1answer
37 views

What is $1^\omega$?

In Wolfram Mathworld, Ordinal exponentiation $\alpha^\beta$ is defined for limit ordinal $\beta$ as: If $\beta$ is a limit ordinal, then if $\alpha=0$, $\alpha^\beta=0$. If $\alpha\neq 0$ then, ...
0
votes
0answers
31 views

Find all solutions to $a^b = b^a$ [duplicate]

Find all ordered pairs $(a, b)$ such that $a$ and $b$ satisfy $a^b$ = $b^a$ and $a$ and $b$ are integers. The only way I can think of solving this question is by trial and error, but there must ...
1
vote
1answer
46 views

Why is it justified to move the limit into the exponent?

On my last math test my teacher told me that my notation for evaluating limits "might be problematic." The notation he is referring to is when evaluating a limit of the form $$\lim_{x\to ...
0
votes
0answers
13 views

Bounding an expression

I am trying to figure out an upper bound on the following expression $$(1 + \epsilon)^{\frac{A}{1+\epsilon} - B}$$ where $\epsilon \in (0,1)$, $A \in (0,1)$ and $B \in \{0, 1\}$. I tried doing the ...
0
votes
3answers
50 views

Find the value of the expression $\left (2 + 5 \right ) + \left (2^{2}+5^{2} \right ) + \left (2^{3}+5^{3} \right ) + \left (2^{4}+5^{4} \right )$

How to effectively solve this expression? $$\left (2 + 5 \right ) + \left (2^{2}+5^{2} \right ) + \left (2^{3}+5^{3} \right ) + \left (2^{4}+5^{4} \right )$$ Inefficient method: $$\left ...
0
votes
0answers
38 views

What are the solutions to $n^k \equiv k \mod m$?

Question: For a given modulus $m$ and base $n$, I am examining the set of solutions $\{k \in [0, m) \:\: | \:\: n^k \equiv k \mod m\}$. The case when $m$ is a power of 10 interests me the most. Why ...
3
votes
3answers
85 views

Limits using L'Hopital's rule $\lim_{x\to0^+} (x^{x^x-1})$

Could you help me with this one? Thanks. The answer should be 1, somehow. I tried everything I know, but I couldn't solve it. $$\lim_{x\to0^+} (x^{x^x-1})$$
0
votes
2answers
24 views

Exponential to polar form

I have exponential form $$ je^{-j\pi/2} $$, where $j = \sqrt{-1}$ I want to convert this to polar form $$j(\cos\pi/2 + j \sin \pi/2)$$ is it correct?
1
vote
1answer
54 views

Comparing $3^{1431}$ with $2^{2010}$ without logarithms

I want to compare : $3^{1431}$ and $2^{2010}$ I tried logarithms, $\mathrm{log}_2$ is the way to go. $\mathrm{log}_2(2^{2010})=2010$ $\mathrm{log}_2(3^{1431})=1431\,\log_2(3)=2268.08$ since ...
3
votes
2answers
24 views

Exponential Properties

Here are my steps: $e^{{2\pi i}/100} = (e^{\pi i})^{{2/100}} = ((-1)^2)^{1/100} = 1^{1/100} = 1$. I'm not sure if the normal rules of exponents apply like this if the power is complex.
13
votes
1answer
117 views

Numbers whose powers are almost integers

Some real numbers $\alpha$ have the property that their powers get ever closer to being integers -- more precisely, that $$ \lim_{n\to\infty} \alpha^n-[\alpha^n] = 0 $$ where $[\cdot]$ is the ...
-1
votes
2answers
48 views

$x^y = \exp( \ln(x) \cdot y )$, not a real solution for decimal numbers?

I am trying to understand how to calculate $x^y$ where $y$ is a decimal number, ($2^{2.3}$) According to wikipedia, the 'solution' would be $$ x^y = \exp( \ln(x) \cdot y ).$$ But if we break it ...
1
vote
1answer
30 views

Quick check on multiplied powers

I'm feeling a little silly askign this question, but after about 2 hours of circling around the same point I am getting frustrated. Starting with the expressions for $M$ and $R$ from the lecture ...
1
vote
1answer
19 views

Variables and exponents

How would you solve this equation ? $$500n=4000(1.016)^n$$ I tried using some logarithms but I could not do it. The only unknown variable is n but I'm having a bit of trouble getting there.