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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0answers
29 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
93 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
101 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
69 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
50 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
82 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
33 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
26 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
vote
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
vote
3answers
61 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
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3answers
115 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
vote
4answers
68 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 ...
1
vote
2answers
147 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
vote
0answers
16 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
votes
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
126 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
30 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
26 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
26 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
41 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
51 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
81 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
42 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
79 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
40 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
74 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
120 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
43 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
58 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
52 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
41 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
88 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
64 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
25 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.
14
votes
1answer
130 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
50 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
32 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.
8
votes
3answers
118 views

Simplify the expression $(a+1)(a^2+1)(a^4+1)\cdots(a^{32}+1)$

How do I simplify this expression? $$\left( a+1 \right) \left( a^{2}+1 \right) \left( a^{4}+1 \right) \left( a^{8}+1 \right) \left( a^{16}+1 \right) \left( a^{32}+1 \right)$$
1
vote
3answers
251 views

Is there an algorithm to split a number into the sum of powers of 2?

Am I able to split, lets say 76, into the sum of powers of two, through an algorithm and without cycling through possible combinations? For the example above, the answer would be '2^6+2^3+2^2' or ...
0
votes
0answers
13 views

sample size calculation for count data

I have a plan to see some treatment effect in several projects in my company where I will compare the average number of errors now and after the treatment. So what I know from current situation is ...
0
votes
1answer
39 views

Supremum and infimum of powers with rational exponents. [duplicate]

I asked this question earlier and got a very useful hint, but I still can't get the full solution. Here is the original question: I'm trying to show that for $b>1$, $x>0$ and $x$ irrational, ...
1
vote
2answers
42 views

An interesting log problem $3^{{(log_3{x})}^2}+x^{log_3x}=162$

$$3^{{(\log_3{x})}^2}+x^{\log_3x}=162$$ How do I go about doing this. I am stuck at the step $x^{\log_3x} = 81$. Is this right? How do I continue or is it wrong?
3
votes
0answers
37 views

Trying to show that $\sup \{b^p:p\in Q,\;0<p<x\} = \inf\{b^q:q\in Q,\;x<q\}$

I'm trying to show that for $b>1$, $x>0$ and $x$ irrational, that $$\sup \{b^p:p\in Q,\;0<p<x\} = \inf\{b^q:q\in Q,\;x<q\}$$ I know this follows immediately if we define ...
2
votes
0answers
35 views

Exponentiation - I can do this?

Euler showed that: $$ e^{i\pi} + 1 = 0 $$ So I thought: $$ b \in \mathbb R,i = \sqrt{-1} \\ n^{bi} = e^{bi\ln n}\\ b = 2k\pi\\ e^{bi\ln n} = e^{2k\pi i\ln n} = (e^{i\pi})^{2k\ln n} = ((-1)^2)^{k\ln ...