Questions about exponentiation

learn more… | top users | synonyms (1)

0
votes
3answers
63 views

How is $ i^{-1} = -i$ and $i^{-3} = i$?

Now I know that with positive powers of $i$ the cycle is: $i , -1 , -i , 1\ldots$ The negative power cycle is: $-i , -1 , i , 1 \ldots$ Can someone explain to me how $\frac 1 {\sqrt{-1}}$ is equal ...
0
votes
3answers
56 views

modules with several powers $x^{y^z}$

I have an assignment(which gives no credit) which I cant solve so I would like to get some help $3^{{1001}^{1002}}\pmod {43}$ (All original numbers were replaced) Our hint was to use the ...
-5
votes
0answers
67 views

How to solve $121212125^5-121212124^2$ [on hold]

I few hours ago I found this exercise: Find $121212125^5-121212124^2$ I have no idea on how to solve this, please, any help would be appreciated..
1
vote
1answer
84 views

Inequality: $\left|x^3-y^3\right|<|x|^3+|y|^3$

Could anyone show me why $$\left|x^3-y^3\right|<|x|^3+|y|^3$$ for all real numbers (x,y) except 0? I'm thinking of whether of how to remove the modulus sign on the left hand side of the ...
1
vote
2answers
57 views

How does exponentiation relate to multiplication?

My book derives the logarithm function as a definite integral of $1/x$ and defines the exponential function as its inverse. It then extends this definition to other bases: $$b^x = e^{\ln (b) x}$$ ...
3
votes
1answer
68 views

Integral of an exponent of an exponent

For a homework problem, I have to integrate this: $$\int{4^{(4+x)^x}}dx$$ How would I go around to starting this question? I don't know how to evaluate this, and I've tried to use u-subs and ...
3
votes
1answer
92 views

How can I do this? $\int\frac{dx}{x^4+1}$ [duplicate]

I tried to integrate this: $\displaystyle\int \dfrac{dx}{x^4+1}$ I tried to do it with the partial fractions method (after factoring the denominator), but the process is really large, and I got a lot ...
0
votes
2answers
41 views

Matrix Exponent - equivalent of a rotation matrix

For every Rotation Matrix,there is a Matrix Exponent representation where the power is a skew symmetric matrix. More clearly if I have a rotation matrix ${R}_{3 \times 3}$ then there will be a skew ...
6
votes
2answers
208 views

Evaluating a limit. What makes the equality right?

I'm reading a proof of a limit calculation. The limit is: $$\lim\limits_{x\to 0}\left(\frac{a^x+b^x}{2}\right)^\frac{1}{x}$$ where $a,b>0$. The aother claims that: $$\lim\limits_{x\to ...
0
votes
2answers
43 views

When $\ln(1+y) = y + o(y)$?

I was reading a proof which utilize the fact that: $\ln(1+y) = y + o(y)$ http://math.stackexchange.com/a/842557/160028 I'm not so sure what is the meaning of $\ln(1+y) = y + o(y)$. When is it ...
1
vote
3answers
89 views

Does $x^2+ x^4/(3\cdot4) + x^6/(3\cdot4\cdot5\cdot6) + \cdots$ have any compact form?

Is there any compact form for the following series $$F_1(x) = x^2+ \frac{x^4}{3\cdot4} + \frac{x^6}{3\cdot4\cdot5\cdot6} + \cdots$$ $$F_2(x) = x+ \frac{x^3}{2\cdot3} + \frac{x^5}{2\cdot3\cdot4\cdot5} ...
5
votes
3answers
118 views

Solve for $x$ in the equation [closed]

Please help me to solve for x using maybe logarithm or exponential rules (or both) $$ 5^x=2 \cdot 3^x $$
1
vote
3answers
65 views

generalized way of finding pair solutions of an equation

I want to find out pair solutions of this equation: $$x^{2}-79y^{2}=1$$ This is a hyperbola equation. I sketched its graph, but that didn't help me. I think the square from (form?) of $x$ and $y$ is ...
2
votes
2answers
59 views

Evaluate $\frac{\frac{1}{1}}{\frac{1}{5^{-2}}}$

I solved a question in the Manhattan GRE 5 pound book (specifically the 11th question in the Exponents and Roots section). I evaluated $\frac{\frac{1}{1}}{\frac{1}{5^{-2}}}$ as $5^{-2}$ and then ...
2
votes
4answers
450 views

Solving an equation in $x$, in which $x$ occurs as exponent four times

Find the number of solutions to the equation $$2011^x+2012^x+2013^x-2014^x=0$$ The answer seems to be zero, but I have no idea why. Please avoid considering complex solutions and other scary things.
3
votes
2answers
69 views

Solution for this Logarithmic Equation

Recently I was going through a problem from the book Problems in Mathematics - *V Govorov & P Dybov* . $$(x-2)^{\log^2(x-2)+\log(x-2)^5-12}=10^2\log(x-2)$$ I tried solving by first considering ...
2
votes
2answers
67 views

How can I calculate the median value?

$$f(a,b) = a^b$$ Where $0\le a \le1$ and $0\le b \le1$ and either $a\ne0$ or $b\ne0$ How can I calculate the median value of $f$ ? I can estimate it to be about 0.76536 by taking values along the ...
5
votes
3answers
119 views

Prove $\lim_{n\to\infty} \frac{(2^{2^n}+1)(2^{2^n}+3)(2^{2^n}+5)\cdots (2^{2^n+1}+1)}{(2^{2^n})(2^{2^n}+2)(2^{2^n}+4)\cdots (2^{2^n+1})}=\sqrt{2}$

Question: Prove or disprove $$I=\lim_{n\to\infty} \frac{(2^{2^n}+1)(2^{2^n}+3)(2^{2^n}+5)\cdots (2^{2^n+1}+1)}{(2^{2^n})(2^{2^n}+2)(2^{2^n}+4)\cdots (2^{2^n+1})}=\sqrt{2}$$ I know ...
4
votes
0answers
63 views

Definition of $0^z$, $z$ complex

Usually we define the function $a^b$ over the complexes using the exponential function, as $e^{b\log a}$. This function has some issues with multivalued-ness, but it still more or less satisfies ...
3
votes
4answers
101 views

Solving $e^{4x}+3e^{2x}-28=0$

How to solve this equation: $$e^{4x}+3e^{2x}-28=0$$ I don't know how to solve this problem. I read over another example, $e^{2x}-2e^x-8=0,$ and it said that $e^{2x}$ is $e$ to the $x$ squared, ...
0
votes
3answers
42 views

Converting exponents to scientific notation

I have to solve or estimate the answer to an equation that is as follows: $$P_\text{blocks} = \frac{398 \cdot 19^{65}}{\prod^{66}_{i=0} 78804 - i}$$ It doesn't take long to realize that this is an ...
1
vote
1answer
25 views

Transforming a power tower to a product

It is possible to write the product of a sequence of terms $a_i$ as a function of the sum of a sequence of functions of these terms: $$\prod_i a_i=f\left(\sum_i g(a_i)\right)$$ where $f=\exp$ and ...
2
votes
1answer
41 views

Convergence of an infinite power

There are complex numbers $z$ and $w$ for which $$\lim_{n\rightarrow\infty}z\uparrow\uparrow n=w$$ where $\uparrow\uparrow$ is the tetration symbol, e.g. $z=\sqrt{2}$ and $w=2$. Are there complex ...
-2
votes
2answers
62 views

How can I simplify the expression $\frac{\sqrt[5]{x^2}}{x^2}$?

$$\dfrac{\sqrt[5]{x^2}}{x^2}$$ I'm doing a summer math packet for calculus. I need to simplify the above. I think I may know the answer, but I'm not sure. Can someone help me, please?
3
votes
3answers
66 views

$x^y < y^x$ for $y\ll x$?

Sorry if this is a naive question; I am not very good at mathematics. It seems obvious that for many $x$ and $y$, $x^y < y^x$ if $y \ll x$, e.g. $2^{10} > 10^2$. If $x$ and $y$ are very close ...
10
votes
3answers
259 views

Calculate $\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$

I'm an eight-grader and I need help to answer this math problem. Problem: Calculate $$\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$$ This one is very hard for me. It ...
0
votes
1answer
43 views

Why, or why not, is $5^{log_3(n)} \in \mathcal{O}(n^2)$?

Why, or why not, is $5^{\log_3(n)} \in \mathcal{O}(n^2)$ ? I tried transforming the logarithm to a base of 5, so that the logarithm and power cancel each other out. However, when I try to so I get ...
1
vote
1answer
82 views

combination of quadratic and cubic series

I'm an eight-grader and I need help to answer this math problem (homework). Problem: Calculate $$\frac{1^2+2^2+3^2+4^2+...+1000^2}{1^3+2^3+3^3+4^3+...+1000^3}$$ Attempt: I know how to calculate ...
1
vote
1answer
68 views

Square root of $\frac{2}{2^x}$; how do I find $x$?

I have this: $$\sqrt{\frac{2}{2^x}} = 9.313225746154785 \times 10^{−10}$$ (sqrt(2/(2^x))) How should I find $x$? I know it's 61 for this case, but I'd like to know how to solve it for when I don't. ...
3
votes
1answer
39 views

Calculating limit in parts. Why possible?

Let $f$, continuous function, differentiable at $x=1$ and $f(1)>0$. Consider the following equation: $$\lim \limits_{x\to 1} ...
1
vote
1answer
39 views

How to show that $w$ is a $N$th primitive root of unity?

I am studying the discrete Fourier transform. For sequence $x_{0}, \dots, x_{N-1}$ it is defined as $$X_{k} = \sum_{n=0}^{N-1} x_{n}e^{-2\pi ikn/N} \quad 0 \leq k \leq N-1$$ according to Wikipedia. ...
2
votes
4answers
120 views

Proving exponential is growing faster than polynomial

Let $P(x)$, a polynomial which isn't the zero-polynomial. I want to prove the following limits $$\lim \limits_{x\to\infty} \left|P(x)\right|e^{-x} = 0$$ $$\lim \limits_{x\to-\infty} ...
2
votes
2answers
89 views

Is $\lim_{x\to 0+} \frac{\ln(x)}{\ln(x)} = \frac{-\infty}{-\infty} = 1$

$$\lim_{x\to 0^+} \sin(x)^\frac{1}{\ln(x)} = ... = \exp \left(\lim_{x\to 0^+} \frac{\ln(\frac{\sin x}{x}) + \ln(x)}{\ln(x)}\right)$$ Now, from continuity we can evaluate each term separately. ...
3
votes
2answers
49 views

Find the value of $\frac{S_{5}S_{2}}{S_{7}}$

If $a$, $b$, $c$ $\in \mathbb R$, we define $S_{k}=\frac{a^k+b^k+c^k}{k}$ (where $k$ is a non-negative integer). Given that $S_{1}=0$, find the value of $$\frac{S_{5}S_{2}}{S_{7}}$$ I tried: ...
8
votes
5answers
935 views

Solving base e equation

So I ran into some confusion while doing this problem, and I won't bore you with the details, but it comes down to trying to solve $e^x - e^{-x} = 0$. I know to solve it, we can rewrite it as $e^x - ...
1
vote
0answers
47 views

$U(n) =$ the smallest strictly positive integer $k$ such that $2^k-1$ is divisible by the first $n$ primes. What are the first values of $U(n)$?

$U(n) =$ the smallest strictly positive integer $k$ such that $2^k-1$ is divisible by the first $n$ prime numbers (except for the first prime number: $2$). What are the first values of $U(n)$ up to ...
0
votes
1answer
52 views

How do they integrate this exponential?

Below, I tired to integrate te^(-j2pi*t) from 0 to 1. But am not getting what my professor got for n not equal to zero, which is also shown. I tried LIATE but am always getting something with an ...
2
votes
4answers
113 views

$(-1)^{0.2}=0.8090 + 0.5878i$ how can this be?

I'm working on a numerical analysis project (working with matlab a lot) and I noticed that when I ask for matlab to compute the exponent of a negative number, it gives wrong output when the exponent ...
2
votes
2answers
26 views

Concerning Rules of Exponents & Absolute Value

I understand that one of the accepted definitions of the absolute value function is $\left| x \right| = \sqrt{x^2}$. However, I do not understand why if I substitute $-5$ in for $x$ that I can't do ...
4
votes
2answers
160 views

Combination of quadratic and arithmetic series

Problem: Calculate $\dfrac{1^2+2^2+3^2+4^2+\cdots+23333330^2}{1+2+3+4+\cdots+23333330}$. Attempt: I know the denominator is arithmetic series and equals ...
0
votes
0answers
56 views

Exponential integral with $x^2$ and $\cos x$

The first part is just a Gaussian integral and the second is the modified Bessel function of the first kind for $n=0$, but I couldn't find any information and what to do with their summation. Any tips ...
0
votes
2answers
19 views

Find numbers who compose the sum of sequence numbers(nth root)

I have an sequence of numbers 1, 2, 4, 8, 16, 32 and so on. Given an number, which must be the sum of the sequence, for example: 44(which is the sum of 4+8+32). How do I know the number 44 is composed ...
0
votes
1answer
30 views

Algebraic Proof of Sum of Exponential Powers is Product of Exponentials

Can somebody provide a proof of the summation of powers law for the product of two exponentials, using only algebra and the Taylor series, no derivatives or calculus tricks?
1
vote
1answer
49 views

Why do we define the complex exponential as we do?

Why do we define the complex exponential as we do? Defining $e^{z}$ as $e^{x}e^{iy}$ certainly seems to make sense, but I'm not sure the formal reason as to why it's defined like this. Was it from ...
2
votes
3answers
89 views

Solve exponential equation $3^x= 2^x+2$

How do we solve this? I can't think of an easy way.. Is there any way to solve it without using newton's method or other approximations? $3^x=2^x+2$
1
vote
0answers
18 views

Fast way to find exponential of a matrix dot product where one of them is diagonal

Suppose $Q$ is a dot product of diagonal matrix A and matrix B: $$ Q=A\cdot B= \left( \begin{matrix} a_1 & 0 & \cdots & 0 \\ 0 & a_2 & \cdots & 0 \\ ...
1
vote
4answers
73 views

closed-form term for this sum:

related to this question: Is there an easy closed-form term for $$\sum_{j=k}^{\infty} \frac{x^j}{j!}e^{-x},$$ thus when the sum starts at a constant $k$ instead of $1$? EDIT: Thanks for your help. ...
8
votes
2answers
71 views

If any integer to the power of $x$ is integer, must $x$ be integer?

My apologies if this has been asked already, I've searched but couldn't find it... Let $x$ such that for every $y \in N$, $y^x$ is an integer. Does that necessarily mean that $x$ is an integer?
2
votes
1answer
187 views

Solving Weird Exponential Equations

I am working on my math homework when I encountered a difficult problem. I simplified the equation and substituted smaller numbers to get this: $n*2^n>10$ I have tried standard algebraic ...
0
votes
1answer
40 views

Choosing a branch of the square root

Assume $O$ is the compliment of the non-positive part of the real line to the complex plane. This is an open and connected set. Only one of the values of $\sqrt z$ in $O$ has positive real part. With ...