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2
votes
5answers
1k views
Why is two to the power of zero equal to binary one?
Probably a simple question and possibly not asked very well. What I want to know is..
In binary, a decimal value of 1 is also 1.
It can be expressed as $x = 1 \times 2^0$
Question:
Why is two to ...
2
votes
5answers
302 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#?
15
votes
0answers
197 views
+200
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 ...
5
votes
5answers
304 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
33 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
$$ \sqrt[w]{z} = z^\frac{1}{w} \qquad ...
1
vote
4answers
63 views
Complex power of a complex number: Find $x$ and $y$ in $x + yi = (a + bi)^{c+di}$
$$
x + yi = (a + bi)^{c+di}
$$
Find $x$ and $y$ in terms of $a$, $b$, $c$ and $d$.
Where, $i$ is defined as $\sqrt{-1}$ and $a$, $b$, $c$, $d$ are real numbers.
I defined two new real number ...
7
votes
1answer
138 views
Is it possible to prove the positive root of the equation ${^4}x=2$, $x=1.4466014324…$ is irrational?
(somewhat related to my earlier question)
Let ${^n}a$ denote tetration $\underbrace{a^{a^{.^{.^{.^a}}}}}_{n \text{ times}}$ (or, defined recursively, ${^1}a=a$, ${^{n+1}}a=a^{({^n}a)}$).
The ...
2
votes
1answer
65 views
Definite integral including the ratio and power functions of a single variable
I find trouble in calculating the following integral:
$$
\int_0^R \frac{m\cdot x}{m+s\cdot x^a} \,dx
$$
Mathematica does not provide an output for this function, however, there seems to be an output ...
1
vote
2answers
58 views
Trace of the matrix power
Say I have matrix $A = \begin{bmatrix}
a & 0 & -c\\
0 & b & 0\\
-c & 0 & a
\end{bmatrix}$.
What is matrix trace
tr(A^200)
Thanks much!
4
votes
4answers
57 views
Can you raise a Matrix to a non integer number? [duplicate]
So I heard you can take a matrix A to the power 2, take it to a -3th power and multiply it by an irrational number. You can also do some other non-intuitive things like taking e to the power of a ...
0
votes
1answer
38 views
Does $ \ (g^a Mod\ p)^b\, $ $\equiv$ $ \ (g^a)^b (Mod\ p)\, $ hold true?
Are these two equations:
$$
\ (g^a Mod\ p)^b\,
$$
$$
\ (g^a)^b (Mod\ p)\,
$$
one and the same? If yes then how And if no then how to solve the first equation?
4
votes
2answers
93 views
How to evaluate powers of powers (i.e. $2^3^4$) in absence of parentheses?
If you look at $2^{3^4}$, what is the expected result? Should it be read as $2^{(3^4)}$ or $(2^3)^4$? Normally I would use parentheses to make the meaning clear, but if none are shown, what would you ...
-1
votes
1answer
55 views
$n^0 = 1$ ? Try for this case? [duplicate]
We know that anything to the power $0 = 1$ i.e. $n^0 = 1$
My question is that, is $0^0 = 0$ or $1$ and why?
9
votes
7answers
632 views
Pattern to last three digits of power of $3$?
I'm wondering if there is a pattern to the last three digits of a a power of $3$? I need to find out the last three digits of $3^{27}$, without a calculator.
I've tried to find a pattern but can not ...
2
votes
4answers
100 views
If $3^x \bmod 7 = 5$, what is $x$ and how?
I am an amateur java programmer who is stuck on this problem:
$$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 ...
