Questions about exponentiation

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0
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0answers
12 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 ...
7
votes
5answers
255 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 $n^0 = 1$.
10
votes
1answer
158 views

Is $\exp:\overline{\mathbb{M}}_n\to\mathbb{M}_n$ injective?

More specific to my problem, this is a variation on Is $\exp:\mathbb{M_n}\to\mathbb{M_n}$ injective? which was promptly answered with a counterexample. Let $\mathbb{M}_n$ be the space of $n\times n$ ...
0
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2answers
41 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 ...
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4answers
31 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 ...
1
vote
2answers
19 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?
2
votes
2answers
41 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
4answers
18 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 ...
12
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6answers
14k views

How to calculate a decimal power of a number

I wish to calculate a power like $$2.14 ^ {2.14}$$ When I ask my calculator to do it, I just get an answer, but I want to see the calculation. So my question is, how to calculate this with a pen, ...
0
votes
2answers
28 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 ...
0
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2answers
41 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 ...
1
vote
2answers
42 views

Matrix Exponential of Identity Matrix

I was just wondering what would the sum be of $e^{I_n}$ where $I_n$ is the identity matrix. I know the maclaurin series for $e^x$ is $1+\frac x{1!}+\frac {x^2}{2!}+...$. I know that $e^0$ is 1 right? ...
0
votes
3answers
38 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
35 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
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3answers
48 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
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2answers
27 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 ...
3
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2answers
61 views

Proving that not defined value is equal to something

My younger brother (9th Grader) got the following maths problem- Given: $$2^a = 3^b = 6^c$$ Prove: $$c=\frac{a * b}{a+b}$$ From my elementary knowledge of mathematics it seems like a=b=c=0.Also, ...
1
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3answers
36 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
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1answer
54 views

another question on surds and how to use math symbols in this site

$$\sqrt{ 3x }= x + \sqrt {3}$$ this is what i tried $$\sqrt{ x }= (x + \sqrt {3})^2\\ = x^2 + 3 $$ Give x in the form $$A \sqrt {B} + C $$ Can you show me how this is done step by step. The ...
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
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2answers
39 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
628 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
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2answers
27 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
34 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 ...
6
votes
1answer
181 views

Exponential of a function times derivative

Exponential of a derivative $e^{a\partial}$ is simply a shift operator, i.e. \begin{equation} e^{a\partial}f(x)=f(a+x) \end{equation} This can be easily verified from a Taylor series \begin{equation} ...
0
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0answers
21 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 ...
94
votes
9answers
3k views

What does $2^x$ really mean when $x$ is not an integer?

We all know that $2^5$ means $2\times 2\times 2\times 2\times 2 = 32$, but what does $2^\pi$ mean? How is it possible to calculate that without using a calculator? I am really curious about this, so ...
0
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0answers
25 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, ...
84
votes
14answers
6k views

Zero to the zero power - Is $0^0=1$?

Could someone provide me with good explanation of why $0^0 = 1$? My train of thought: $x > 0$ $0^x = 0^{x-0} = 0^x/0^0$, so $0^0 = 0^x/0^x = ?$ Possible answers: $0^0 * 0^x = 1 * 0^x$, so ...
0
votes
0answers
39 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
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1answer
45 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
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0answers
29 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 ...
3
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3answers
215 views

Comparing $\large 3^{3^{3^3}}$, googol, googolplex

How to show that $\large 3^{3^{3^3}}$ is larger than a googol ($\large 10^{100}$) but smaller than googoplex ($\large 10^{10^{100}}$). Thanks much in advance!!!
2
votes
2answers
119 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
51 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 ...
11
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6answers
1k views

Matrix to power $2012$

How to calculate $A^{2012}$? $A = \left[\begin{array}{ccc}3&-1&-2\\2&0&-2\\2&-1&-1\end{array}\right]$ How can one calculate this? It must be tricky or something, cause there ...
0
votes
2answers
48 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 ...
1
vote
1answer
26 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 ...
1
vote
1answer
18 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\} ...
0
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1answer
34 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 ...
12
votes
9answers
4k views

How to calculate $e^x$ with a standard calculator

Is there a simple method for calculating the $e^x$ ($x\in\mathbb{R}$) with a basic add/subtract/multiply/divide calculator that converges in reasonable time, preferably without having to memorize ...
3
votes
1answer
44 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
26 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 ...
7
votes
4answers
559 views

How to evaluate fractional tetrations?

Recently I've come across 'tetration' in my studies of math, and I've become intrigued how they can be evaluated when the "tetration number" is not whole. For those who do not know, tetrations are the ...
1
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1answer
41 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 ...
0
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0answers
9 views

Find the inverse function of a function relating to limited exponential sum

The function is given out as: $$y = 4x + {x^m} + {x^{ - m}},where{\text{ 0 < m}} \leqslant {\text{1, 0 < }}y < 6;$$ Closed form will be highly appreciate,but approximate results is also ...
0
votes
0answers
23 views

How can I simplify the following expression with exponents.

$$\frac{(t+1)^{\frac{1}{3}}-\frac{1}{3}t(t^2+1)^{-\frac{2}{3}}}{(t^2+1)^{-\frac{2}{3}}}$$ I found this problem from a book and its answer is $\frac{2t+3}{3(t+1)^{\frac{4}{3}}}$(as in the book's ...
0
votes
1answer
16 views

Name of numbers in “to the power of” and factorial calculations

In $4*5=20$ , $4$ and $5$ are multiplicands and $20$ is the product. What are the names / labels of the numbers in the following expressions? $2^3=8$ $4!=24$
0
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2answers
33 views

How to solve an exponential function with multiple addends

Our math teacher gave us the following exponential equation to solve: $3^x+10=2*7^x$ ...and I was stumped. Eventually, the solution given was to graph both sides and find their intersection using a ...
19
votes
8answers
2k views

How does the exponent of a function effect the result?

The $x^{2/2}$ can be represented by these ways: $$\begin{align} x^{2\over2}=\sqrt{x^2} = |x|\\ \end{align} $$ And $$\begin{align} x^{2\over2}=x^{1} = x\\ \end{align} $$ Which one is correct? And what ...