Questions about exponentiation

learn more… | top users | synonyms (1)

0
votes
2answers
22 views

Exponential equation help

Solve $x^{ln x} = e^{(lnx)^{3}}$ I'm looking at the mark scheme but I don't understand what they've done. I'd appreciate it if someone could explain every step. MS: taking ln of both sides or ...
2
votes
0answers
36 views

Powers of (large) lower triangular matrix

Consider the following "game" of chance. Each time the player pushes a button he is awarded a random (finite, integer, non-negative) number of points. The probability of receiving any particular score ...
1
vote
4answers
80 views

How to find the integral $\int4^{-x}dx$?

What approach would be ideal in finding the integral $\int4^{-x}dx$?
5
votes
2answers
84 views

Generalized power rule for derivatives

Background This background is not really necessary to answer my question, but I included it here to provide context. This question has some programming aspects to it as well, but since my question ...
0
votes
1answer
19 views

Variable Base with Variable as Factor in Exponent, Find Value

I saw a problem recently that looked like this: Assume $w$ and $z$ are positive. If $z^{4w} = 64$, what does $z^{6w}$ equal? And I had absolutely no idea how to even begin attempting this equation. ...
1
vote
3answers
42 views

How to find exponent of a number in a combination?

How do I find the exponent of $7$ in $^{100}C_{50}$ that is, $\dfrac{100!}{(100-50)!\cdot 50!} =\dfrac{100!}{50!\cdot 50!}$, this question was out of the blue, and I haven't been able to find any ...
0
votes
2answers
47 views

rational exponents. two differing answers.

This is not homework. Example 3) (d) of section P.4, rational exponents in Algebra and Trigonometry: $$\frac{1}{\sqrt[3]{x^4}} = \frac{1}{x^\frac43} = x^{-4/3}$$ Completely rational. Almost ...
1
vote
1answer
44 views

Is there a convention for power of a half being the positive square root?

I know the $\surd$ sign refers to the positive square root. Does the exponent 1/2 mean the positive square root too by convention? I ask because I'm converting from parametric to cartesian here... ...
0
votes
1answer
44 views

Homework help to rearrange formula

Given the equation $${V_m} = u(\ln {m_0} - \ln {m_8}) - g{t_f}$$ I need to solve for ${m_0}$ Here is what I have but it looks messy and I feel like there is sometihng wrong or a better way 1st ...
1
vote
2answers
39 views

Reciprocal of $7.5^{1-x}$

Ok my calculator tells me the reciprocal of $7.5^{1-x}$ is $0.1333\cdot7.5^x$. Can anyone explain the steps involved to get this manually? Is it along the line of the reciprocal of $7.5^1 + 7.5^{-x} ...
1
vote
2answers
30 views

$3^a\mid s(n) \Rightarrow 3^a\mid n$

This is not a homework question, neither a championship problem (as far as I've searched in the net), and it came up noticing a singular pattern, involving the powers of $3$: "Prove or disprove that ...
1
vote
1answer
28 views

How does $10^{100}$ = $2^{\frac{100}{\log2}}$?

Googol is equal to $10^{100}$. To determine the number of bits that it needs to represented in binary, we need to rewrite Googol with a base of $2$. This is the correct answer: $$10^{100} = ...
1
vote
0answers
45 views

Justification for exponents other than positive integers

Here's a question that's bothered me ever since highschool, and I've never heard a good answer. I know that mathematicians can define operators to mean whatever they want, as long as their system of ...
1
vote
2answers
39 views

where does the modulus go when cancelling $e$ and $\ln$ in this problem?

So I did this problem today: Show that $\frac{dy}{dx} = yx^2$ can be written as $y = Ae^{\frac{x^3}{3}}$ my solution is shown below: $$ \frac{dy}{dx} = yx^2 $$ $$ \frac{1}{y} dy = x^2 dx $$ $$ ...
3
votes
1answer
71 views

On the equation $\exp(a x+b)=\ln(x)$

I am confronted with: $$\exp(a x+b)=\ln(x)$$ for $a,b$ reals and $a<0$, $b>0$. I need the (unique) solution for $x$. My first target is (if it exists) an analytic solution in terms of ...
0
votes
2answers
23 views

Find the largest exponent

I've got this GRE math question: The integer y is positive. If $6^y$ is a factor of $(2^{14})(3^{24})$, then what is the greatest possible value of y? The answer is ...
2
votes
0answers
21 views

Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $Dq(x) . Ax < 0$ for all $x \neq 0$

Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $$Dq(x) . Ax < 0$$ for all $x \neq 0$ Definition: a linear system $x' = Ax$ called ...
1
vote
1answer
64 views

Conditions required for $(z_{1}z_{2})^{\omega}=z_{1}^{\omega}z_{2}^{\omega}$, where $z_{1},z_{2},\omega\in\mathbb{C}$

I am having trouble finding the conditions on $z_{1}$ and $z_{2}$ in order for: $$(z_{1}z_{2})^{\omega}\equiv z_{1}^{\omega}z_{2}^{\omega}\qquad \forall\omega\in\mathbb{C}$$ My first step was to ...
8
votes
5answers
290 views

Why does $n^0 = 1$?

Why is it that $n^0 = 1$? I understand how $n^2 = n*n$ and how $n^1 = n$ but I can't understand why $n^0 = 1$.
1
vote
4answers
38 views

Find $m$ and $n$

Two finite sets have m and n elements. Thew total number of subsets of the first set is 56 more than the two total number of subsets of the second set. Find the value of $m$ and $n$. The equation ...
0
votes
2answers
50 views

Simple question on exponentiation

I know this one is trivial, but I was wondering: if I have something like $$a^{b^c}$$ then i know that it should be read as $$a^{\left(b^c\right)}$$ if no other parenthesis is present. Question: if ...
1
vote
2answers
22 views

Difference between growth formulas

What is the difference between $$N = N_0 \cdot e^{kt}$$ and $$N= N_0(1+r)^n$$ I'm trying to find the best formula to calculate population growth and sources seem to vary between these two?
0
votes
4answers
27 views

solve for x, giving answer to 3s.f?

I need help solving the question below: $$ 2x^ \frac{1}{4} = \frac {64} {x} $$ I know the answer is 16 but I'm not sure how to get to it. Can you explain how to get the answer so I can solve similar ...
0
votes
2answers
32 views

What is the 'growth constant'?

I'm looking into the formula of growth, namely $$N= N_0 e^{kt}$$ where $k$ is the 'growth constant'. What is the growth constant and how do I find it? I'm looking at a bug that has on average 1,67 ...
2
votes
2answers
66 views

compute the integral using residue theory

I am trying to compute an integral in an example in my complex analysis textbook: $$\int_{-\infty}^\infty {xsinx\over x^4+1}dx$$ The book gives some startup hints, but I don't quite follow, I set ...
0
votes
2answers
58 views

Is there a simple algorithm for exponentiating large numbers to large powers?

I've been thinking about this for some days, a multiplication is a lot of sums, so: $$75\times 75=\overbrace{75+75+75+75+75+75+75+75+\cdots}^{\text{75 times}}$$ But then, there is a simple algorithm ...
0
votes
3answers
48 views

Error raising a complex number to a power

I am trying to do $(3+7i)^5$ which acording to WolframAlpha and Mathway should be: $23028−11228i$ Yet I instead get: $6123+14287i$ -- I'm getting that answer by doing: $3^5 ...
2
votes
2answers
44 views

magnitude of complex exponental always equals 1?

as we all know $$e^{j\theta} = \cos\theta + j\sin\theta \\ |e^{j\theta}| = \sqrt{\cos^2\theta + \sin^2\theta} = 1$$ That means $|e^{j\theta}| = 1$ with any value $\theta$ is ($2\pi, \frac{\pi}{3}$, ...
2
votes
3answers
57 views

Finding the matrix exponential

Find the matrix exponential of $$\begin{bmatrix}1& 1\\ 0& 1\end{bmatrix}.$$ Since this matrix is not diagonalizable, you will have to use the definition of the matrix exponential. ...
0
votes
2answers
31 views

Power calculation for simplification?

I have this simple question I saw here: ±(2 - 2^(-23)) × 2^128 = ±6.8 × 10^38 How did they get to ...
1
vote
3answers
41 views

question on surds i already asked this question but the answer I got did not match the one in the book [duplicate]

$$\sqrt{ 3x }= x + \sqrt {3}$$ Give x in the form $$A \sqrt {B} + C $$ Can you show me how this is done step by step. The answer I have in the book is: $$\frac {1}{2} \sqrt{3} + \frac {3}{2} $$ ...
0
votes
2answers
50 views

If $\sqrt{x+y}+\sqrt{y+z}=\sqrt{x+z}$, then $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=?$

If $\sqrt{x+y}+\sqrt{y+z}=\sqrt{x+z}$, then $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=?$ I really am stumped on this problem. I squared the first equation and found that $-y = \sqrt{(x+y)(y+z)}$. So ...
4
votes
3answers
699 views

how to calculate 2^1.4

So I have got a very basic question but it didn't come up as a google search so I am posting it here. I want to know how to easy calculate 2^1.4 = 2.6390... ...
0
votes
2answers
37 views

Subtracting 2 fractions with variables in the denominator that have different exponents.

Sorry for the relatively elementary question, but I am having trouble remembering exactly how to do this type of problem. I am looking to simplify this: $$ \frac{3}{4t^{1/4}} - \frac{1}{2t^{3/4}} $$ ...
1
vote
2answers
40 views

Inequality with trigonometric functions

Find all values for $a$ such that the following inequality holds: $$\sin^6x + \cos^6x + a\sin x \cos x \ge 0$$ To be fair, I didn't manage to get anything helpful wiht my calculations. I tried to ...
0
votes
0answers
31 views

What is the general notation for the principal value of complex exponential?

It is general to distinguish the principal value of complex logarithm set by denoting it $Ln( z)$. Is there any general notation to distinguish the principal value of complex exponential? In complex ...
0
votes
0answers
27 views

Function plotting

I have a function $f(x)=\binom{N}{K} \ln(1-F(x)), x \geq 0$, where $F(x)$ is a cumulative distribution function. Then, $\ln(1-F(x))$ is negative for various values of $x$ as $F(x) \geq 0$. Also, ...
1
vote
1answer
58 views

Does a solution exist where $p,q$ are odd primes and $p^a - q^b = p^c - q^d$ where $a > c > 1$ and $b > d > 1$

From my thinking so far, there is no solution. Is this an open question or is the answer well known? Here's my reasoning about this issue: If a solution exists, then: $$p^c(p^{a-c} - 1) = ...
0
votes
0answers
42 views

Solving equation for powers

I would like to find $\gamma$ in: $$ \sum_{i=0}^n x_i^\gamma = y $$ where $n$, $0 \leq x_i \leq 1$ and $0 \leq y \leq n$ are known. Also, $n$ can be fairly large (i.e. from a few thousands to a few ...
1
vote
1answer
67 views

What does it mean when a matrix is to the (-1/2) power?

I'm reading a machine learning paper that uses a form of matrix normalization called symmetric divisive; given a matrix A and a diagonal matrix D derived from A, we define $$N=D^{-1/2}AD^{-1/2}$$ I am ...
0
votes
0answers
34 views

Find $a,b,c \ge 2$ and $p,q$ odd primes where $p^a - 1 = c*q^b$

I've been recently thinking about finding primes $p,q$ where the power of one divides the power of the other when subtracted by $1$. For example, if we remove the requirement that $p,q$ be odd ...
2
votes
2answers
124 views

First derivative of multiplied powers

Wolfram Alfa shows $\frac{d}{dx}e^{4y} = 4e^{4y}$ but I do not understand how to get to that answer I have $e^{4y} = (e^4)^y$ So by the chain rule is it not the case that \begin{align} ...
1
vote
2answers
65 views

Exponential of a 3x3 lower bidiagonal matrix

I have a 3x3 matrix with non-zero entries ONLY along the main diagonal and the diagonal above. There are exactly two non zero diagonals in the matrix like this \begin{pmatrix} a & 0 & 0 \\ d ...
1
vote
1answer
20 views

Supremum of (e^(i z t) - 1)/z

i'm new here, so i'm not sure if this is the right place to ask this question: I know that the following holds true: $$ \forall\, t \in \mathbb{R} \; \forall\,x\in\mathbb{R}\setminus\{0\} ...
1
vote
1answer
45 views

weighted average with exponential weighting

I want to create weighted average, where weights depend on value of number. If I want exponential weights is this regular? $average = \log_e(\frac{\sum_{i=1}^n e^{v_i}}{n})$ Isn't it just average of ...
0
votes
2answers
53 views

complex expression to the power of a complex expression

I have a math exam tomorrow, and i am not sure with my solution for a exercise. can you please tell me if i am right. Question is: $$(1+i)^{(1-i)}$$ My solution is: $$\sqrt{2} e^{(i ...
0
votes
1answer
37 views

Why expression under root has to be positive?

I have function defined like this : f(x,y) = $\sqrt[127,5]{\frac{x^²+y^²-4y}{4x-x^2-y^2}}$ I thouth that domain is $4x-x^2-y^2 \neq 0$ but when I looked on wolfram, the domain is everything under the ...
3
votes
1answer
48 views

matrix exponential limit

I'm having litlle trouble here to prove the following statement: "Let $A$ an $n\times n$ matrix (real or complex). Prove that $$\lim_{n \to \infty} \left(I + \frac{A}{n}\right)^{n} = e^{A}.$$ Now ...
0
votes
1answer
34 views

Why are exponents not associative?

I ran into something that seemed odd to me today: exponents are not associative. The following equation sums that up: $$ 10 * 2^{5x} \not\equiv 20^{5x} $$ Why is this the case? Is there some ...
1
vote
1answer
56 views

Cauchy's integral formula used on circle

If $\gamma$ is a piecewise, smooth, positively oriented simple closed curve in $D$, then Cauchy's formula states that $f(z)=1/2\pi i\int_\gamma {f(a)\over {a-z}}$. My textbook also stated that for ...