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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Are there natural numbers $x$, $y$, $z$, such that $x^4+y^4-2^{3z}=3$?
Are there natural numbers $x$, $y$, $z$, such that $x^4+y^4-2^{3z}=3$? Justify!!
I literally have no idea for this problem. I thought of doing the last digit but it doesnt help at all. I wonder if we ...
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How to find rational power of two using Newton's method?
I know how to find integer powers of two, $2^x = \prod_{i=1}^x 2$, I have memorized powers of two up to 32nd power of two, and I use bit-shifts to calculate them. For integer powers of other numbers I ...
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Why doesn't $x^0$ equal the same thing as $x\cdot0$?
If $x\cdot0$ means adding zero $x$s together and $x^0$ means multiplying zero $x$s together then conceptually why aren't they both equal to the same thing?
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What is this constant $\lim_{a\to +\infty}(a-f(u)-\ln(a))=C$?
Problem :
Let the function on $x\in(0,1),a=\operatorname{constant}>1$ :
$$f(x)=x^{a^{x^{a}}}a^{x^{a^{x}}}-\operatorname{arctanh}(x)$$
Now let :
$$f'(u)=0,0.99<u<1$$
Then it seems we have :
$$\...
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Why doesn't this Taylor series for exponentiation work?
Don't know if this is more mathematics or programming. Anyway I want to implement an efficient way to approximate exponentiation involving non-integral exponents, faster than C++'s pow from cmath.
I ...
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Answer $9^x = 4^x + 6^x$ to a 10th grader, who knows math until the equation of a straight line (just before calculus) [closed]
Find $x$:
$$9^x = 4^x + 6^x$$
This was in my exam today, and I have no idea, would really help if someone taught me, I would love to know how to write math on this website and is there a way to like ...
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Which definition of "power" is true: Britannica's or Wikipedia's? [closed]
Britannica says $a^k$ is the power; Wikipedia says that the power is $k$. Which one is it? Did someone make a mistake?
Exponents
Just as a repeated sum $a + a + \cdots + a$ of $k$ summands is written ...
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Solve $a^{2x}+ a^4 = a^{x+1}+ a^{x+3}$
Solve for the values of $x:$
$$a^{2x}+ a^4 = a^{x+1}+ a^{x+3}.$$
My attempt has been to make the base the same so I can cancel and add exponents:
$a^x.a^x+ a^4 = a^x.a^1 + a^x.a^3$
$a^x.a^x+ a^4 = a^x ...
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Create an algorithm to find powers using only addition and one factorial.
Why there is this kind of relation between power and factorial?
I know that there is a way to use this pattern to generate powers using just addition and one factorial, but I can't wrap my puny brain ...
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How would you go about finding the exponent for any given x that most closely matches or surpasses factorial growth?
While analyzing the factorial function and comparing it to basic exponentiation, I couldn't help but notice the obvious fact that exponentiation can eventually overtake factorialization if the ...
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Why is $2^3 = 8$, but $2^{6/2}$ is technically $\pm 8$?
So I'm just going through some basic algebra as a refresher, and I'm wondering why a number taken to a certain power (such as $2^3$) has an additional negative value if the exponent is expressed as a ...
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What is it called when you set $x$ as the index of an root or the base of a log? [closed]
Here are some names of functions (where $x$ is a variable and $a$ is a constant):
Power: $x^{a}$
Exponential: $a^{x}$
Logarithmic: $\log_{a}x$
$N$th root: $\sqrt[a]{x}$
But what is it called if, say, ...
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About Non-integer Exponentiation and Graphs of Exponential Functions
I’m rather new in exploring mathematical subjects in some more detail and this has been one of the questions I haven’t been able to figure out on my own. How exactly do we know that graphs of ...
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How to $a^b \to b^a$ [duplicate]
Addition, multiplication and power are formed in a similar way. One follows from the other by repeating the preceding action several times. But while $a+b=b+a$, and $a \times b=b \times a$, in turn $a^...
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How to evaluate the infinite product $\prod_{n\geq1}\prod_{k\geq1}e^{\frac{(-1)^{n+k}}{n^2k^2+3nk^2+3n^2k+2k^2+2n^2+9kn+6k+6n+4}}$
So I was bored, and decided to do some infinite products and sums for fun. After a while, I came up with this:$$\prod_{n\geq1}\prod_{k\geq1}e^{\frac{(-1)^{n+k}}{n^2k^2+3nk^2+3n^2k+2k^2+2n^2+9kn+6k+6n+...
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Simply calculate $\frac{ \frac{65536!}{(65536 - 100)! }}{65536^{100}}$
I want to calculate this value exactly: $C=\frac{ \frac{65536!}{(65536 - 100)! }}{65536^{100}}$. Or at least I want to obtain an a convincing lower bound to argue that this value is close to 1.
It can ...
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What is the smallest positive integer that can not be written as a sum of positive integer powers of distinct elements of {1,…,n}? [closed]
Lets say you have numbers from 1 to 3, how many numbers can be made using powers of the numbers without the base repeating?
$1=1^1$
$2=2^1$
$3=3^1$
$4=3^1+1^1$
$5=3^1+2^1$
$6=3^1+2^1+1^1$
$7=3^1+2^2$
$...
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Simplification - Power of diagonal matrix and circulant matrix [closed]
I am solving a matrix equation for the elements of the diagonal matrix $D$. Therefore, I must find vector real vector $e$, where $D = diag(e)$.
Vector $x$ and $y$ are given and are complex vectors. V ...
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If $a,b,x\in\mathbb{R}^+$ and $a<b$ then $a^x<b^x$
I am reading in a real analysis text about the definition of exponentiation when the exponent is real.
I have at my disposal all the properties of natural/integer/rational exponentiation and the ...
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Is there a way to simplify this exponent?
I have the exponents $k^4+k^5+k^6$. Is there any way to simplify this into one exponent? I'm trying to find a way to simplify a sequence of increasing exponents into one, but am not sure how. With ...
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Is this a unique property of power functions?
I want to find all continuous functions $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that
$$\forall s \in \mathbb{R}^+: \forall x \in \mathbb{R}^+: \frac{f(x)}{f(s x)} = \frac{f(1)}{f(s)}$$
I know ...
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If $a,b,c$ are nonzero natural numbers that verify the relation $a^2+b^2=c^2$, show that $a^n+b^n<c^n$. [duplicate]
PROBLEM
If $a,b,c$ are nonzero natural numbers that verify the relation $a^2+b^2=c^2$, show that $a^n+b^n<c^n$, where $n$ is a strictly natural number greater than $2$.
WHAT I THOUGHT OF
$a^2+b^2=...
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help me with idea to understand: Determine the real number x for which the value of the expression $E(x)=\sqrt{x^2-4x+5}+\sqrt{x^2-6x+13}$ is minimum
PROBLEM
Determine the real number x for which the value of the expression $E(x)=\sqrt{x^2-4x+5}+\sqrt{x^2-6x+13}$ is minimum.
WHAT I THOUGHT OF
So for $E(x)$ to be the minimum, we have to make $\sqrt{...
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Solve the equation in integers numbers $3x^2-2xy-7=y^2$.
PROBLEM:
Solve the equation in integers numbers $3x^2-2xy-7=y^2$.
WHAT I THOUGHT OF:
What if we pass $y^2$ to the other member of the equality.
$3x^2-2xy-7-y^2=0$
Forward it would be simple if we ...
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Find x and y for $11^x=5*2^y+1$
PROBLEM
Solve the equation in the set of natural numbers:
$11^x=5*2^y+1$
WHAT I THOUGHT OF
We can write $11^x$ as $(10+1)^x$
We know that $(a+b)^x= M_a+b^x=M_b+a^x$
Applying the formula above we can ...
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Is there any way express $2^{6/x} - 2^{x+1} + 12 = 0$ taking $2^x$ as some variable, say $a$?
I tried raising to $x$ on both sides, getting $2^6 - 2^{x(x+1)} + 12^x = 0$, but we still can't simplify the $2^{x(x+1)}$ in terms of $2^x$. Is there any method of simplifying the equation, taking $2^...
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Solve $^y{x} =$ $^x{y}$ over the real numbers
Let $x, y \in \mathbb{R}^{+}$ be such that $x \neq y$ and assume $n \in \mathbb{N}-\{0\}$.
Now, referring to the well-known hyperoperation sequence $x[n]y$, we have that $x[1]y=x+y$ and we know that ...
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Finding a real closed-form solution to a tricky transcendental equation
One of my friends posed the following question with a prize of free bubble tea for anyone who could find a closed-form solution to the problem (not necessarily in terms of elementary functions):
What ...
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Convergence of Variable Base Power Tower $\frac{1}{2}^{\frac{1}{3}^{\frac{1}{4}^{\ldots}}}$
I’m curious if there is a heuristic method that can be used to solve what appears to be an elementary power tower where every increasing power is decreasing by a half integer. Numerical methods ...
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What are negative exponents? How are they used in real life? How does $a^{-b} = 1/a^b$? How does it work in detail? [closed]
I am very confused on what negative exponents are, how they're used in real life, how the formula $a^{-b}$ is the same thing as $1/a^b$, and how this formula works. An example I saw was $10^{-3}$:
$$...
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Do I understand negative exponents correctly?
I know that each time you go up an exponent, you multiply by the base again e.g. $2^2$ to $2^3$. To go down, you you divide by the base. The way I find negative exponents is that I divide 1 by the ...
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How to express power law as a differential equation, while not being a function of absolute time?
The data I have can nicely be described by a power law
$$y=a*t^{-b} + c$$
I need to come up with a differential equation, which the above equation is a solution to. This can be expressed as
$$\frac{dy}...
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Bounding exponential function by n^{-\gamma}
Assume that we know
$$\exp(-\frac{t^2/2}{np(1-p)+t/3})$$
and our goal is to obtain which value of $t$ we can take such that the above is bounded by $n^{-\gamma}$, for certain constant $\gamma $.
The ...
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Why does the function $e^{ix}$ have a real part, without using the Euler's formula
I would like to intuitively understand why $e^{ix}$ has a real part, if the the function $e^{ix}$ has an imaginary argument.
I know that
$$e^{ix}=\cos x + i\sin x$$
and I don't need convincing that it ...
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Modular tetration (power tower) for non-coprime numbers case
I'm writing algorithm for calculating $a^{a^{...^{a}}}$ mod $m$.
According to Euler's theorem, $a^k = a^{k\mod{\phi(m)}}$ mod $m$, if $a$ and $m$ are relatively primes. If $m = \prod_{i=0}^n p_i^{\...
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How fast are the Taylor series of $\ln(1+x)$ and $\exp(x)$ converging to satisfy the $n$th decimal place? Is it dependend on the input $x$?
How fast are the Taylor series of $\ln(1+x)$ and $\exp(x)$ converging to satisfy the $n$th decimal place? Is it dependend on the input $x$?
$x \in \mathbb R+$.
Maybe you can give an $O(n)$ for that - ...
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Why does the exponent rule [If $a^b = a^c,$ then $b=c$] not apply to imaginary numbers [duplicate]
So I came across this video: https://www.youtube.com/watch?v=R476CTKUIr4
in which the creator shows an incorrect proof of π = 0 and the mistake made. The video proves this using the exponent rule $(a^...
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Solve for $x$: $3^{2x}=5^{x-1}$
So I recently got myself an AP Precalculus book (for anyone who wants to know which book, I will have that in the "To Clarify" section at the bottom of this post) and was looking through ...
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Domain of $\frac{1}{x^{-1}}$
Why WolframAlpha and other online calculators compute $x\in \mathbb{R}-\{0\}$ (and not $\mathbb{R}$) as the domain of $\frac{1}{x^{-1}}$?
Since $\frac{1}{x^{-1}}=\frac{1}{\frac{1}{x}}=x$, is not the ...
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Are there any non-trivial solutions of $a^n = b^m + 1$ for natural $a$, $b$ and $n, m \ge 2$
There is an obvious solution $3^2 = 2^3 + 1$.
However, I tried to search for solutions in the range $a, b \in [2,100]$ and $\min(n,m) \in [2,1000]$, but did not find any.
It is easy to show that $a$ ...
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What is the resulting precision from an exponent operation?
Say I want to perform exponentiation on some decimal number (like a measured weight) and some integer exponent (like ^2 or ^3). What are the general rules for the resulting precision(significant ...
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Problem with exponents and inequalities
Hello I tried to solve this problem, below:
Given the three numbers $x,y=x^x,z=x^{x^x}$ with $.9<x<1.0$. Arranged
in order of increasing magnitude, they are:
$\text{(A) } x,z,y\quad \text{(B) } ...
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How to prove that the exponential of an initial object by an object with global elements is iso to the initial object?
I am self-studying out of McLarty's 'Elementary Categories, Elementary Toposes' and I am having trouble proving that if $\oslash$ is the intial object, and $A$ is any element in a cartesian closed ...
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How can I transfer number with exponent to other side of equation without losing its sign?
I just begun learning some math and I stumbled upon this issue.. I am currently programming a little game, where I need to deal with intersection of circle and a line and this is where I need standard ...
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What does graphical solution really mean?
Given question is "Sketch the graph of $y=ln(2x-1)$. Determine the equation of the straight line which would need to be drawn on the same axes as graph of $y=ln(2x-1)$ in order to obtain a ...
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The power of a power of a real number [closed]
I wonder that if $a$ is a positive real number and $b,c$ are real numbers, is it true that $$(a^b)^c = a^{bc}?$$
I only need this fact to prove a nonconstructive problem, so I didn't spend much time ...
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About numbers to the powers of imaginary numbers
I once saw a problem which asks for the real and imaginary parts $(i^i)^i$. The problem also implied that there are several solutions to this. However I am very confused, $(i^i)$ is a real number so ...
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What is $ \lim\limits_{x\to\infty}(\int_0^x e^{t^2}dt)^{\frac{1}{x^2}} $ ？ [duplicate]
$ \lim\limits_{x\to\infty}(\int_0^x e^{t^2}dt)^{\frac{1}{x^2}} \\$
my idea :
$$e^{\lim\limits_{x\to\infty}\ln(\int_0^x e^{t^2}dt)^{\tfrac{1}{x^2}}}=e^{\lim\limits_{x\to\infty}\frac{\ln(\int_0^x e^{t^2}...
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Taking the geometric derivative of $e^{\frac{1}{2}\left[\gamma_0 \vec{k} \vec{x} \gamma^0 - \vec{x} \gamma^0 \gamma_0 \vec{k}\right]}$
As in the title: I'm looking to take the geometric derivative with respect to $\vec{x}$ of the exponential of the commutator of $\gamma_0 \vec{k}$ and $\vec{x} \gamma^0$:
$$e^{\frac{1}{2}\left[\...
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Are there useful applications for the differences between the one-digits of (2^n) - (n^2), where n is a positive integer? 2 can be a different integer
I got curious and started listing the differences between 2^n and n^2. For example, 2^6 = 64, and 6^2 = 36, so the difference is 28. I wanted to see if any relationship could be found and I saw that ...
