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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. ...
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 ...
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: $\left(\frac{\log_{10}(a)}{\log_{10}(2)}\right)\cdot b$ ...
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.
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.
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 ...
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 $\sqrt{1}=-1$ $1=-1$
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 ...
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 ...
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#?
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 ...
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 ...
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 ...
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 ...
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!
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?
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 ...
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?
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 ...
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 ...
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 ...
4answers
72 views

Square and square root and negative numbers [duplicate]

Are they equal? -5 = $\sqrt{(-5)^2}$
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, ...
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 ...
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} & ...
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 ...
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 ...
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 ...
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.
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 ...
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 ...
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 = ...
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?
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 ...
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 ...
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 ...
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!!!
1answer
62 views

How do I solve this (multiplication of exponents)

I can not find any help to spread the exponent across the brackets $$\left[-3(-2)^2\right]^{-1}$$
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
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
0answers
100 views

What is the relationship between these expression?

Moderator Note: This is a Project Euler question If ...
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 ...
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 ...
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 ...
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 ...
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