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

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3
votes
0answers
30 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
34 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 ...
0
votes
4answers
57 views

Why is $\left(\frac{1}{2}\right)^{x} = \frac{1}{7}$ the same as saying: $(2)^{x} = 7$

Why is $\left(\frac{1}{2}\right)^{x} = \frac{1}{7}$ the same as saying: $(2)^{x} = 7$ Sorry for the really dumb question but I'd like to see the process of how this is achieved.
1
vote
0answers
63 views

When is $0^0 = 1$ inconvenient? I heard sometimes $0^0 = 0$ may help.

Don't get me wrong: my question is NOT claiming $0^0 = 1$. I understand it's indeterminate. Many articles show that defining $0^0 = 1$ is (just) convenient and I completely agree. However, I've heard ...
0
votes
1answer
24 views

Existance of a nonempty subset $X$ of $\mathbb{Z}$: $\forall x \in X, k \in \mathbb{N}, \exists ! y \in X$ satisfying $\mid x-y \mid 2^k$

Define the set $A = \{2^k \mid k \in \mathbb{Z}_+\}$ Does there exist a nonempty subset $X$ of $\mathbb{Z}$ satisfying the following condition: For all $a \in A$ and $x \in X$, there exists a unique ...
-2
votes
2answers
48 views

finding the second smallest number [closed]

Please tell me how to solve this kind of problem in a fast manner. Which of the following is the second smallest number: $2^{120}$, $3^{80}$ and $10^{30}$? Hope you could show me the best solution, ...
4
votes
2answers
125 views

Which general physical transformation to the number space does exponentiation represent?

Addition and multiplication may be defined in two ways, one specific and one general: Addition specific: addition is repeated incrementation. This is specific and sub-optimal as while $2 + 4$ is ...
4
votes
3answers
48 views

The value of the cubic root of $-i$

So this was the question given to us. $\left(\iota=\sqrt{-1}\right)$ Value(s) of $\left(-\iota\right)^{\dfrac{1}{3}}$ are (A) $\dfrac{\sqrt{3}-\iota}{2}$ (B) $\dfrac{\sqrt{3}+\iota}{2}$ ...
0
votes
2answers
40 views

If $a^{(b^c)}=d^c$, find $d$ in terms of $a$ and $b$.

Is it possible to express $d$ in terms of $a$ and $b$ only in the following equation? $$a^{b^c}=a^{(b^c)}=d^c$$ I want something like $d=\dots$ Thanks in advance!
1
vote
1answer
63 views

Exponential Factorial vs Tetration

I'm wondering whether there's a known way to compare the exponential factorial of n versus the tetration of a fixed number $($ e.g., $3$, since it appears in Graham's number $)$ with the same number ...
1
vote
0answers
20 views

Solving an Equation Involving Modular Exponation

Is it possible to solve the following equation? $$ 3^k\mod k \equiv 24 $$ Clearly $k>24$; does a solution definitely exist, and can it be found in any simple way? I can certainly calculate ...
2
votes
2answers
21 views

Find real constants $c$ and $k$ such that $y=cx^k$ passes through point $(a, b)$ with slope $m$

In the Cartesian plane, can a power function of the form $y=cx^k$ (where $c>0$ and $k>1$, not necessarily an integer) be found such that its graph passes through any arbitrary point $(a, b)$ ...
0
votes
2answers
34 views

Negative exponents on a quantity in scientific notation considering significant figures

Are there rules that apply to negative exponents with regard to scientific notation? The specific problem is: $$\left(6.3\times10^{2}\right)^{-6}$$ I believe the following is correct: ...
1
vote
1answer
43 views

Convert $3^n$ to some form of $2^n$

I am not from Maths field, but I need your help to convert the $3^n$ form to $2^n$. I need to change the base from $3$ to $2$. The resultant expression can be of any form.
4
votes
1answer
68 views

Why is $\gcd(2^p + 1,3^p + 1) = 1$?

Let $p$ be an odd prime. Why is $\gcd(2^p + 1,3^p + 1) = 1$ ? I tried using fermat's little and $\gcd(a+b,a) = gcd(a,b)$ but without succes. I can make a statistical argument that suggests there are ...
3
votes
1answer
49 views

Number of integer triples to exponential equation

I'm taking a class on number theory and this is one of the problems my professor gave. How many ordered integer triples $(x,y,z)$ are there such that $x^y-y^x=2017\times z$, where $x,y$ are less than ...
0
votes
1answer
41 views

Find the number of times we can take log base two of this exponent

I'm not sure how to simplify this type of exponential expression. I would like to know $k$ many $log_{2}^{k}(n)$ such that $n \leq 1$ $$n = (2^{2^{2^{15}}})^2$$ So attempt I to simplify $$n = ...
0
votes
1answer
34 views

Generalized Complex Exponentiation

Is there a way to create a general formula for exponentiating two complex numbers like there is for addition and multiplication? Ex: $(a + bi) + (c + di) = (a + c) + (b + d)i $ $(a + bi) * (c + di) ...
1
vote
2answers
124 views

Simplifying algebraic fraction, exponents

Would someone be able to tell me how $$\bigg( \frac{5}{a^4} \bigg)^{-3}$$ gets simplified to $$\frac{a^{12}}{125}?$$ Thank you!
0
votes
1answer
19 views

How to simplify algebraic expression with exponents?

Can someone please tell me step-by-step how to simply this? $$\sqrt{\frac{8x^{{\frac{1}{2}}^{\frac{2}{3}}}}{x^{-\frac{1}{2}}}}$$ Edit: correct answer is 2x^(5/12), I'm just not sure how to do it. ...
1
vote
4answers
80 views

How do you prove that for all real numbers c, there always exists a value n, in which $4^n > c3^n$?

What we have so far, its $\frac{4^n}{ 3^n} > c,$ such that $\left(\frac{4}{3}\right) ^ n > c.$ From that I know that it is true, but I don't really know how to prove it formally.
2
votes
2answers
78 views

I am not sure, but it's an equation…

$$2x + 2^x = 4$$ So it's clear that $x$ will equal $1$ but how can it be solved through an algebraic method to be able to determine $x$ with more complex numbers. I've tried to transform the exponent ...
2
votes
3answers
75 views

How to find the value of this expression?

I just saw this question in one exam. Please help me solve it. I am not able to find any clue on where to begin. (ignore that tick it might be wrong)
2
votes
1answer
28 views

How can you write “10 to the power of X” ($10^X$) if you can only use standard text?

Basically the title says it all: You have to fill in a form which only allows standard text in ASCII characters - how would you write $10^X$ then? It should be universally understandable and not be ...
0
votes
0answers
17 views

Fundamental matrix with nilpotent power

Given that a square $n\times n$ matrix $\bf{A}$ has only one eigenvalue $r$ with multiplicity $n$, we have by Cayley-Hamilton Theorem that $(\bf{A}$$-r$$\bf{I})^n$=$\bf{0}$. We can then express ...
0
votes
1answer
31 views

What number/function/thing to the power n gives n itself

Is there a number/function/thing if put to the power n gives n? I know that there is log to the base 2 of 2n gives n. which is log2(2n)=n But i am asking for (thing)n=n. In fact i want an x=thing that ...
3
votes
1answer
84 views

Sum of series: $1^1 + 2^2 + 3^3 + … + n^n$

Searched every where on web but I couldn't find out the formula for this series. That's why I am asking here. I tried following following formula: $((n*(n+1))/2)^n$
0
votes
0answers
41 views

x raised to the power of x raised to the power of logarithm base 3 x

Have a question that has me a little stumped. I know that this equation:$$x^\left(log_3x + x^\left(log_3x\right)\right) \neq 162$$ Simplifies to: $$x^\left(log_3x\right) \neq 81$$ My question is how ...
1
vote
2answers
43 views

matrix exponential property

Take $\alpha\in\mathrm{C}$ and $A$ a complex matrix. Is is true that : $$\exp(\alpha A) = (\exp{A})^\alpha$$ My intuition tells me that this is true. But I can't prove it and I can't find this ...
0
votes
1answer
97 views

Infinite tetration convergence

I came across infinite tetrations on wikipedia (https://en.wikipedia.org/wiki/Tetration) it says that the infinite tetration converges if and only if $\ e^{-e} \leq x \leq e^{1/e}$. I was wondering if ...
0
votes
3answers
22 views

Exponentials with three variables: solving for an equation

$2^{x}=5^{y}=100^{z}$ Find $z$ in terms of $y$ and $x$. The term $z$ should be a function of $x$ and $y$, i.e.: $z(x,y)$. All I could get were recursive attempts.
1
vote
2answers
42 views

How would I calculate sum of digits in the number (a^b)?

I was doing a question from a site,project euler specifically.I came to a question in which I was asked to calculate sum of digits in number 2^1000.Since I program very often I was able to do that ...
1
vote
2answers
40 views

How to Find the Derivatives of $\tan^2(x^4)$ and $\sec^3(x^5)$?

I am to find the derivative of f(x) and g(x): So far, I know the following: The derivative of tan(x) = sec(x)^2 The derivative of sec(x) = sec(x)tan(x) So, I have tried the following steps to ...
1
vote
2answers
71 views

Exponentiating a matrix

I was just wondering whether my solution is correct or not and if it isn't, where I went wrong? Find $ e^A $ where $ A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} $ So what I ...
0
votes
2answers
34 views

Negative number squared in expression $-5-8^{2}$

I know this probably is a silly stupid question, but I just don't get it. I'm currently doing Khan Academy pre-algebra and stumbled upon an awkward problem. I assume that: $-5-8^{2}=59$ Because -8 ...
1
vote
1answer
38 views

Raise an inequality to power

I'm looking for general rules for working with inequalities and powers. Are those general rules correct? 1.You may raise any inequality to an odd power of N - and keep the inequality as it was. For ...
0
votes
1answer
98 views

Radioactive Decay formula is $A=A_0e^{-kt}$. How many years until 10 grams decay so that only 8 remain

I have been trying this question for hours and come to a dead end every time... Consider the radioactive decay formula $A=A_0e^{-kt}$ where $A$ is the amount of radium remaining at the time $t$. ...
1
vote
3answers
34 views

Floating point overflow. Can this power equation be simplified?

I’m running into trouble with the following formula that we’re using in our software. $$ \frac{1 - x^{-y}}{1 - x^{-(y+1)}} $$ In certain cases, the value for $y$ is a relatively large number; ...
0
votes
2answers
48 views

Math Team Problem Involving Powers of Powers of 3

So I am in my high school math team and I was given the following expression $$3^{3^{3^{...}}}$$ Where there are multiple powers of 3 with a total of two thousand and fifteen 3's. The question ...
0
votes
0answers
38 views

Curve of product of 2 powers

could anyone help me identify the name of the green curve in the attached image, please? It's the product of the two other curves, which are power (to the -6) in blue; and area of a circle (which I ...
0
votes
1answer
54 views

How do you calculate the module of a high raised number?

I need some help solving this. Find $3^{336549354297854} (mod 28)$ I don't really understand how to solve this type of problem. can anyone explain?
1
vote
3answers
40 views

How to solve exponential expressions?

There are some questions that I have. Question 1) $$ (x^2/5)^3 = 2^6/5^y$$ To find the $y$ I used the same base $$ 1/5^3 = 1/5y$$ Teacher told that the exponent will be the same if equaled, so $ ...
3
votes
3answers
57 views

Solve exponential equation with exponent variables.

Ok, the question is: Solve the following for x: $2^x4^{x-1}=70$ I have just asked Wolfram Alpha, of course though, it supplies an answer without revealing its working. I started to try the ...
1
vote
0answers
39 views

Efficient way to find primitive root without prime factorization

I was just wondering if there is a more efficient bruteforcing approach to find any primitive root of number p without prime factorization. My approach is as ...
0
votes
1answer
28 views

How to find coefficients after raising a power series to certain positive integer power?

Suppose a power series $S=\sum_{n=0}^\infty a_nx^n$. How do we find the coefficients for $S^k = (\sum_{n=0}^\infty a_nx^n)^k = \sum_{n=0}^\infty c_nx^n$. Is there a convenient way to write $c_n$ in ...
6
votes
3answers
103 views

When is $(a^b)^c $ = $a^{bc}$ true?

I know that in some cases this rule is not true. For example $$((-1)^2)^\frac{1}{2}\ne(-1)^{(2\cdot\frac{1}{2})}$$ So when is this rule true ?
3
votes
2answers
92 views

$2\cdot5^x-7^x-4^x>0$ for $-1\le x<0$

Show that $2\cdot5^x-7^x-4^x>0$ for $-1\le x<0$. I tried by differentiation but I found it useless as expression become more complicated. Also tried by Jensen's inequality but did not ...
0
votes
2answers
21 views

$x^a$ not defined in $\Re$ when x negative?

I know that by definition $x^a$ = exp(alogx) which is not defined when x not in $\Re^∗_+$ Does it mean that for example $-1^3$ not defined for real numbers ?
1
vote
2answers
84 views

Find root of $x\cdot 2^x - 1$ function

Is it possible to solve the equation: $x2^x - 1 = 0$ without using the function graph? According to the function plot, its root is around 0.6. I need to get the numeric value of the function's ...
0
votes
4answers
78 views

How to solve $(\frac{1}{x})^2x=e^6$? [closed]

I need to solve this: $$\left(\frac{1}{x}\right)^2x=e^6$$ I know it's equal $$\ x=\frac{1}{e^6}$$ But how?