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

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0
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1answer
27 views

Exponentiating roots of unity

When given some number some complex $2n^{th}$ root of unity, $z$, how does one evaluate something such as $z^{2m}$ (with $m=kn$)? I would take $z^{2m}=(z^{2n})^k=1^k=1$, but I don't know if this is ...
2
votes
1answer
36 views

Unable to process Large numbers [closed]

A small spherical cell of diameter $1.616E^{-35}$ is exponentially multiplying as $2^n$ where n is the generation number. The duration of 1 generation is $5.39E^{-44}$ second. And the cells cluster ...
2
votes
1answer
71 views

How to calculate 10^ decimal power without a calculator?

I need to know how to calculate 10^ a decimal power, like 10^-7.4, without a calculator, in as simple a way as possible, since I will be doing questions which only allow me about a minute to a minute ...
1
vote
0answers
26 views

-1 raised to fractional indices lying between 0 and 1

For a personal project, I had to figure out what happens when $-1$ (negative one) is raised to fractional powers lying between $0$ and $1$. I thought that if I get a power $x = 0.a_1a_2a_3...a_n$ (...
8
votes
7answers
1k views

What is the right way to calculate a power?

I noticed that there are two solutions for $(-1)^{14/2}$: $((-1)^{14})^{1/2} = 1$ $(-1)^{14/2}=(-1)^7=-1$ What am I doing wrong?
5
votes
2answers
505 views

What exactly are those “two irrational numbers” $x$ and $y$ such that $x^y$ is rational? [duplicate]

It's possible to prove nonconstructively that there exists irrational numbers $x$ and $y$ such that $x^y$ is rational, but that proof only proves that such numbers exist and does not specify what they ...
1
vote
0answers
50 views

Got stuck while integrating $\int x^x dx$ [duplicate]

What is the integration of $$\int x^x dx$$ And how can I understand whether an integration is possible or not? Is there any rule to understand whether a function is integrable or not?
0
votes
3answers
44 views

SAT question about integers and exponents

If $a$ and $b$ are positive integers and $$(a^\frac{1}{2}\times b^\frac{1}{3})^6=432$$ What is the value of $ab$?
0
votes
3answers
90 views

Finding the function that would describe this:

I'm not going to go into detail why I am interested in the next iteration of these functions, but here they are: 1: 6/(x+1) 2: 8/(2^x) 3: 10/(?) The question is, which one is next? I will say that ...
2
votes
0answers
48 views

How to define matrix power

I am currently writing a scriptum struggling with the definition of matrix power. Precisely, let $ \mathbb A \in \mathbb C^{n, n} $ and $ p \in \mathbb R $. I Currently have: If $ p \in \mathbb N $ ...
2
votes
3answers
78 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 \mathbb{...
2
votes
3answers
120 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
23 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
72 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
81 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
38 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: $\sum_{i=1}^k n_i^2 + 2\sum_{i=1}^{n-1} \sum_{j=i+1}^{n}n_in_j$...
1
vote
2answers
61 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
vote
0answers
30 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 ...
-7
votes
2answers
98 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
votes
5answers
105 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
70 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
91 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
34 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 edx....
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
66 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
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 $f:\mathbb{R_+...
1
vote
4answers
72 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
161 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
17 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 2%....
0
votes
2answers
22 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
127 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
32 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
124 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
29 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
54 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: $$\left(\...
2
votes
1answer
83 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
43 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
80 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
42 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
123 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: $$\lim_{n\to+\...
3
votes
1answer
45 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 be a ...
1
vote
1answer
68 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 \infty}{f(x)}^...
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 ...