Questions about exponentiation

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2
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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
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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
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3answers
40 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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2answers
49 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
686 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
35 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
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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
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0answers
29 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
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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
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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) = ...
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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
62 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
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
38 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
52 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
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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
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1answer
30 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
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1answer
55 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 ...
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0answers
11 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
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0answers
30 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
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1answer
22 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? $2^3=8$ $4!=24$
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2answers
35 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 ...
6
votes
1answer
232 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} ...
21
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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 ...
0
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1answer
48 views

Why 1.66x10e-27 is 1.66e-26

I am confused with this new way of writing exponential. For example I saw a question where person has changed 1.66X10e-27 to 1.66e-26. I am confused how -27 is reduced to -26 thanks
3
votes
1answer
79 views

Is there any way to analyze an absurdly large exponent?

On a recent Giant Bombcast, someone wrote in and asked an absurd question (as is usual for this podcast). In short, the question was: Given a 1080p TV, how long would it take to view every ...
2
votes
1answer
35 views

Permuting digits in a power of $2$

Does there exist a natural number $N$ that is a power of $2$ whose digits (in the decimal representation) can be permuted to a different power of $2$? Thoughts: If such a number $N$ exists, then ...
3
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2answers
71 views

Exponents with the same power

I've wanted to practice solving simple operations on exponents, so I've made a couple of equations to which I know the answers. $$5^x -4^x = 9$$ I feel really stupid, because I can't solve this one ...
0
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0answers
28 views

Trigonometric Power Formulas (or something more modest)

How does one begin to show (natural $n$): $$\cos^{2n}(x) =\frac{1}{2^{2n}} \binom{2n}{n}+ \frac{1}{2^{2n-1}} \sum_{k=0}^{n-1} \binom{2n}{k} \cos[2(n-k)x]$$ $$\cos^{2n+1}(x) =\frac{1}{4^{n}} ...
13
votes
5answers
427 views

Value of $(-1)^x$ for $x$ irrational

I was working on an analysis problem when this question arose in one my proofs. I think it may be either $-1$ or $1$, but it seems like there can only be an arbitrary way to assign this. So is there ...
0
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3answers
65 views

$(5^{2x}-1)(5^x)=1/5^x$ solve

I have the problem $(5^{2x}-1)(5^x) = 1/5^x$. I have already simplified it to $5^{3x}-1=1/5^x$ My question is when I do $\log$ base $5$ to the left side of the equation to get $3x-1$ by itself so ...
1
vote
1answer
48 views

moving trig function from the exponent?

Is there a form of $e^{\sin(x)}$ that does not have trig functions in the exponent? I've seen the "Euler Identity" form and looked into series expansions. (thanks for the answers -- sorry I didn't ...
0
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0answers
63 views

Complex exponentiation

So I've got this question that is a bit difficult to ask, since it uses a term in my language that I can't properly translate into English. For $z\in\mathbb{C}^*$ and $a\in\mathbb{C}$ it would be ...
4
votes
2answers
106 views

$2 \uparrow^n 2 = 4$ and the magnificence of $2$

I was reading up on tetration when I realized: $$2 \uparrow\uparrow 2 = 2\uparrow 2 = 2 \times 2 = 2+2 =4$$ Infact, when generally speaking: $$ 2 \uparrow^n 2 =4$$ Now, I realize that this is because ...
0
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3answers
69 views

Intricate exponential equation

This is the question: $$ \frac{(2^{3n+4})(8^{2n})(4^{n+1})}{(2^{n+5})(4^{8+n})} = 2 $$ I've tried several times but I can't get the answer by working out.I know $n =2$, can someone please give me some ...
0
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0answers
15 views

How would I design a formula that increases the length of pauses exponentially based on current speed?

I'm writing a program that presents users words in a flash-card fashion, at a speed they define (say, 500 cards/min). When a "section" of cards is done, I want there to be a pause before the next one ...
-2
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1answer
100 views

why e raised to power 1 is 2.71.. [closed]

Please can somebody explain and interpret why e raised to power 1 is 2.71 and from where did it come around in mathematics? since it has become vital in mathematics and science.
0
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1answer
19 views

Simplify this indices?

Simplify this: $6a^3 * {a^{-5}\over2}$ I got $6a^3 * {1\over2a^5}$ What should I do next? please explain with steps.
13
votes
1answer
183 views

How to raise a number to a quaternion power

Now, I know that it's (relatively) easy to calculate, say, $r^{a+bi}$ (using the fact that, for $z_1, z_2\in \mathbb{C}, {z_1}^{z_2}=e^{z_2\ln(z_1)}$ and $\ln(z_1$) can just be found using: ...
0
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1answer
72 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 ...
0
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1answer
34 views

Need help with self study on surds

give x in the form A $ \sqrt B$ + C $(\sqrt 3x) = 3$ can someone show me how to solve this please?
41
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10answers
2k views

What is exponentiation?

Is there an intuitive definition of exponentiation? In elementary school, we learned that $$ a^b = a \cdot a \cdot a \cdot a \cdots (b\ \textrm{ times}) $$ where $b$ is an integer. Then later on ...
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2answers
80 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? ...
1
vote
3answers
55 views

If $(x^y)^z = x^{y\cdot z}$, why does $(-5)^{2^{0.5}}$not equal $(-5)^1$?

As shown by Wolfram Alpha, $(x^2)^{0.5}$ is equal to |x|, but if you tried to simplify it to $x^{2\times {0.5}}$, it would just be $x^1$, or $x$. Is there some unwritten rule about that distribution ...
2
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1answer
54 views

$|x|^{|x|}$ is continuous in $\mathbb{R}$

I'm trying to show this now my self, but still no go. There isn't really a concrete attempt that I can show.. Help?
4
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2answers
129 views

Last digits don't change when exponentiating

While playing around with Wolfram Alpha, I noticed that the last four digits of $7^{7^{7^{7^7}}}, 7^{7^{7^{7^{7^7}}}},$ and $7^{7^{7^{7^{7^{7^7}}}}}$were all $2343$. In fact, the number of sevens did ...