The powers tag has no wiki summary.
2
votes
2answers
27 views
rounding up to nearest square
Say I have x and want to round it up to the nearest square. How might I do that in a constant time manner?
ie.
$2^2$ is 4 and $3^2$ is 9. So I want a formula whereby f(x) = 9 when x is 5, 6, 7 or 8. ...
8
votes
6answers
875 views
10 to the power of 3.5 [closed]
So $10^3 = 10\times 10\times 10 = 1000$, this is really easy to understand.
But what about: $10^{3.5}$ My logic would suggest this was
$10\times 10 \times 10\times 5 = 5000$ but the calculator ...
3
votes
1answer
70 views
taking the log of $a^b$ (Project Euler problem 29)
I've been stuck on Project Euler problem 29 and thus asked a friend who solved it how to do it.
What he basically did was for each power was: $\left(\frac{\log_{10}(a)}{\log_{10}(2)}\right)\cdot b$ ...
1
vote
1answer
50 views
Solving an equation with $x$ as powers
How would I go about solving $$2^x -2^{x-2}=3 *2^{13}$$Hints please. Thank you.
7
votes
16answers
1k views
Why is $\;n^2-\frac{n^2}{2} =\frac{n^2}{2}\;$? [closed]
Could someone please expand on how to get from $\;\displaystyle\left( n^2-\frac{n^2}{2}\right)\;$ to $\;\left(\dfrac{n^2}{2}\right)\;?\;$
I can't seem to wrap my head around that.
2
votes
3answers
19 views
why if x in 1/n power >(<) y in 1/m power then x in c/n power >(<) y in c/m power?
As you might guess this is one more stupid question from non-matematician, and you are right. I found this exercise in "Algebra and trigonometry book":
$7^{1/2}$ or $4^{1/4}$. After some googling I ...
6
votes
6answers
355 views
How is this proof flawed?
$\sqrt{x}=-1$
$\sqrt{x}^2=(-1)^2$
$x=1$
Now substitute it into the original equation
$\sqrt{1}=-1$
$1=-1$
1
vote
1answer
35 views
Can we write $\sqrt[w]{z}=z^\frac{1}{w}$ when both $w$ and $z$ are complex numbers? [duplicate]
Let $w$ and $z$ be complex numbers defined in terms of real numbers $a$, $b$, $c$ and $d$ as follows:
$$ w = a+bi \\ z = c+di $$
Can we analogically write
$$ \sqrt[w]{z} = z^\frac{1}{w} \qquad ...
8
votes
1answer
210 views
Is it possible to prove the positive root of the equation ${^4}x=2$, $x=1.4466014324…$ is irrational?
(somewhat related to my earlier question)
Let ${^n}a$ denote tetration $\underbrace{a^{a^{.^{.^{.^a}}}}}_{n \text{ times}}$ (or, defined recursively, ${^1}a=a$, ${^{n+1}}a=a^{({^n}a)}$).
The ...
2
votes
5answers
412 views
What is the value of $2^{3000}$
What is the value of $2^{3000}$?
How to calculate it using a programming language like C#?
24
votes
1answer
368 views
How do I calculate the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$
I need to find the 2nd term of continued fraction for the power tower ${^5}e=e^{e^{e^{e^{e}}}}$ ( i.e. $\lfloor\{e^{e^{e^{e^{e}}}}\}^{-1}\rfloor$), or even higher towers. The number is too big to ...
2
votes
1answer
71 views
Definite integral including the ratio and power functions of a single variable
I find trouble in calculating the following integral:
$$
\int_0^R \frac{m\cdot x}{m+s\cdot x^a} \,dx
$$
Mathematica does not provide an output for this function, however, there seems to be an output ...
1
vote
4answers
70 views
Complex power of a complex number: Find $x$ and $y$ in $x + yi = (a + bi)^{c+di}$
$$
x + yi = (a + bi)^{c+di}
$$
Find $x$ and $y$ in terms of $a$, $b$, $c$ and $d$.
Where, $i$ is defined as $\sqrt{-1}$ and $a$, $b$, $c$, $d$ are real numbers.
I defined two new real number ...
4
votes
4answers
66 views
Can you raise a Matrix to a non integer number? [duplicate]
So I heard you can take a matrix A to the power 2, take it to a -3th power and multiply it by an irrational number. You can also do some other non-intuitive things like taking e to the power of a ...
1
vote
2answers
65 views
Trace of the matrix power
Say I have matrix $A = \begin{bmatrix}
a & 0 & -c\\
0 & b & 0\\
-c & 0 & a
\end{bmatrix}$.
What is matrix trace
tr(A^200)
Thanks much!
0
votes
1answer
42 views
Does $ \ (g^a Mod\ p)^b\, $ $\equiv$ $ \ (g^a)^b (Mod\ p)\, $ hold true?
Are these two equations:
$$
\ (g^a Mod\ p)^b\,
$$
$$
\ (g^a)^b (Mod\ p)\,
$$
one and the same? If yes then how And if no then how to solve the first equation?
4
votes
2answers
96 views
How to evaluate powers of powers (i.e. $2^3^4$) in absence of parentheses?
If you look at $2^{3^4}$, what is the expected result? Should it be read as $2^{(3^4)}$ or $(2^3)^4$? Normally I would use parentheses to make the meaning clear, but if none are shown, what would you ...
-1
votes
1answer
55 views
$n^0 = 1$ ? Try for this case? [duplicate]
We know that anything to the power $0 = 1$ i.e. $n^0 = 1$
My question is that, is $0^0 = 0$ or $1$ and why?
9
votes
7answers
654 views
Pattern to last three digits of power of $3$?
I'm wondering if there is a pattern to the last three digits of a a power of $3$? I need to find out the last three digits of $3^{27}$, without a calculator.
I've tried to find a pattern but can not ...
2
votes
4answers
102 views
If $3^x \bmod 7 = 5$, what is $x$ and how?
I am an amateur java programmer who is stuck on this problem:
$$3^x \bmod 7 = 5$$
then what is $x$ and how? If you can even explain the method for how to arrive at the solution, then it will be very ...
1
vote
0answers
62 views
Are there other powers which follow the rule $a^b = b^a$ than $2^4$? [duplicate]
I was trying to find these powers, but to my disappointment I only found $2^4 = 4^2$. Edit: $a$ must be different to $b$ of course.
Is that the only possible setting, and why? If we assume the number ...
0
votes
4answers
72 views
4
votes
5answers
341 views
Why does the power rule work?
If $$f(x)=x^u$$ then the derivative function will always be $$f'(x)=u*x^{u-1}$$
I've been trying to figure out why that makes sense and I can't quite get there.
I know it can be proven with limits, ...
0
votes
0answers
36 views
Name of odd powered polynomial graph (Opposite of parabola(ic))
I am writing an assignment and have to describe the graphs for when the powers are even and when they are odd. I described the even power graphs as being parabolic or parabolas. The only problem is, I ...
2
votes
1answer
63 views
Why the nth power of a Jordan matrix involves the binomial coefficient?
I've searched a lot for a simple explanation of this. Given a Jordan block $J_k(\lambda)$, its $n$th power is:
$$
J_k(\lambda)^n = \begin{bmatrix}
\lambda^n & \binom{n}{1}\lambda^{n-1} & ...
2
votes
5answers
100 views
Why when indices cancelled out leave 1 at top?
This is a really basic question, but I have just got interested in math and learning rules about powers/indices and this confused me a little.
$\dfrac{a^3}{a^7}$ after they cancel out we get ...
1
vote
2answers
65 views
Calculating new vector positions
I'm using the following formula to calculate the new vector positions for each point selected, I loop through each point selected and get the $(X_i,Y_i,Z_i)$ values, I also get the center values of ...
2
votes
4answers
165 views
1 to the power of infinity, why is it indeterminate? [duplicate]
I've been taught that $1^\infty$ is undetermined case. Why is it so? Isn't $1*1*1...=1$ whatever times you would multiply it? So if you take a limit, say $\lim_{n\to\infty} 1^n$, doesn't it converge ...
2
votes
5answers
68 views
Proof for power functions
Which is greater? $\sqrt{n}^{\sqrt{n+1}}$ or $\sqrt{n+1}^\sqrt{n}$
I know that $\sqrt{n}^{\sqrt{n+1}}$ is greater but I tried using induction and I couldn't figure it out. Thanks for the help.
1
vote
1answer
66 views
Units in exponent - e.g. solve: $2^{3 years}$
What happens to units in an exponent?
My math textbook just introduced the exponential equation:
$$A_t = Pe^{rt}$$
I've always made it a point in solving math problems to include the units in every ...
0
votes
2answers
72 views
Simplifying $ \;x({y^{3}}/{x^{4}})^{1/4}$
I’m a little unsure how to simplify the following expression:
$$
x\left(\frac{y^{3}}{x^{4}}\right)^{1/4}
$$
According to the answer, this should get you $\;\; x y^{3/4} x^{-1} = y^{3/4} $.
My ...
6
votes
5answers
130 views
Solving an equation with fractional powers
I was trying to find the maximum value for a function. I took the first derivative and arrived at this horrible expression:
$$ (x^2 + y^2)^\frac{3}{2} - y {\frac{3}{2}}(x^2 + y^2)^{\frac{1}{2}}2y = ...
2
votes
3answers
66 views
Powers and the logarithm
By example:
$4^{\log_2(n)}$ evaluates to $n^2$
$2^{\log_2(n)}$ evaluates to $n$
What is the rule behind this?
1
vote
2answers
73 views
How to prove that the following language is not regular?
This is the following problem that I've been having difficulty on:
For this problem, we will show that there are non-regular languages over the alphabet $\{0\}$. The language that will be used is the ...
2
votes
4answers
110 views
What is the value of $i^i$? [duplicate]
I understand that when you raise any number $x$ to a power, you multiply $x$ by itself the number of times indicated in the power. However, what happens when $i^i$ is performed? How can a number be ...
1
vote
1answer
59 views
How to expand a fraction in powers of $z$ or $\dfrac{1}{z}$, and which to do, in determining Laurent series
I have a function $f(z)=\dfrac{12}{z(2-z)(1+z)}$, I'm trying to find the Laurent series for each of the three annuli. The singularities are at $z = 0$, $z = 2$, and $z = -1$, so I'm looking for three ...
4
votes
3answers
161 views
Comparing $\large 3^{3^{3^3}}$, googol, googolplex
How to show that $\large 3^{3^{3^3}}$ (Third Ackermann number) is larger than a googol ($\large 10^{100}$) but smaller than googoplex ($\large 10^{10^{100}}$).
Thanks much in advance!!!
0
votes
1answer
62 views
How do I solve this (multiplication of exponents)
I can not find any help to spread the exponent across the brackets
$$\left[-3(-2)^2\right]^{-1}$$
2
votes
4answers
290 views
What is wrong with this problem
We know that:
$(a^n)^m=a^{nm}$
From this we have:
$-3^3=[(-3)^2]^\frac{3}{2}=(3^2)^\frac{3}{2}=27$
Find what's wrong
1
vote
1answer
59 views
Realistic Example of Power-Law Distribution?
I'm missing a bit of inbetween-math, and having some trouble understanding this, but:
I want to generate a set of data that follows a power law. Let's say I have 10,000,000 people who like a ...
4
votes
1answer
127 views
Prime decomposition of an integer: methods of determining the prime factors $ p_1, p_2, …, p_r$ and powers $k_1,k_2, …, k_r$
Any integer n can be written in the form
$ n = p_1^{k_1}p_2^{k_2} ... p_r^{k_r} $,
where the powers $ k_1, k_2, ...,k_r $ are integers and
$ p_1, p_2, ..., p_r$ are primes.
Now I am interested in ...
2
votes
2answers
93 views
Calculating powers
I was thinking how I could program powers into my application. And I want to do this without using the default math libraries. I want to use only +,-,* and /. So I wondered what is the definition of a ...
8
votes
5answers
172 views
Solving an equation with a logarithm in the exponent
I try to solve the following equation:
$$ (N+1)^{\log_N{125}} = 216 $$
I know the answer is 5 here but how could I rewrite the equations so I can solve it?
I tried to take the log of both sides but ...
0
votes
2answers
59 views
How to efficiently compute the coefficients in a bi-binomial expansion?
Is there a computationally efficient way of calculating the coefficients of the polynomial expansion of expressions like $(1+x^a)^m(1+x^b)^n$ for arbitrary positive integers $m,n,a,b$ (and especially ...
0
votes
2answers
103 views
pow$(X,Y)$ $>$ pow$(Y,X)$, if $X<Y$.
How can we proof following?
if $X < Y$, then:
$X^{Y} > Y^{X}$ ,
Where X, and Y are integers. Also $X,Y > 1$.
Except a special case $2^{3} < 3^{2}$.
I think for other ...
0
votes
0answers
100 views
What is the relationship between these expression?
Moderator Note: This is a Project Euler question
If ...
14
votes
2answers
254 views
Prove that for any nonnegative integer n the number $5^{5^{n+1}} + 5^{5 ^n} + 1$ is not prime
My math teacher gave us problems to work on proofs, but this problem has been driving me crazy. I tried to factor or find patterns in the numbers and all I can come up with is that for $n > 0$, the ...
6
votes
3answers
147 views
What's the intuition behind non-integer exponents/powers
Consider some $a \in \mathbb{R}$ and $x \in \mathbb{R}\backslash \mathbb{N}$.
Is there some intuition to be had for the number $a^x$?
For example the intuition of $a^2$ is obvious; it's $a*a$ which ...
3
votes
3answers
95 views
What is the difference between exponentials and powers?
I am a java programmer. But I have a doubt regarding a mathematics. There was a method called Math.exp(double a) description:Returns Euler's number e raised to the power of a double value. and another ...
1
vote
1answer
81 views
Solve for variable inside multiple power in terms of the powers.
I'm a programmer working to write test software. Currently estimates the values it needs with by testing with a brute force algorithm. I'm trying to improve the math behind the software so that I can ...
