Questions about exponentiation, the operation of raising a base $b$ to an exponent $a$ to give $b^a$.

learn more… | top users | synonyms (1)

0
votes
2answers
13 views

How to estimate magnitude of expontent?

When given an exponent, such as 6^12, is there a simple way to approximate how large(magnitude) the result is, without performing the calculation? Is this method accurate for large exponents?
2
votes
2answers
249 views

0's Exponents are impossible? [duplicate]

I've had something that's been bugging me, and I tried research and asked my math teacher. None had sufficient answers. The concept of $0$ is that when $0$ goes to any exponent except for $0$, it ...
-4
votes
1answer
25 views

Need Help: Exponential Equations (Same bases) [on hold]

$3^{n+2} + [3^{n+3} - 3^{n+1}] = ?$ How do we get the answer for this? Do I just remove the bases and proceed to find the value of $n$ or do I use logarithms?
1
vote
2answers
36 views

Exponential function negative: $\left(\frac{81}{4}\right)^{1/4}\left(\frac{1}4\right)^{-3/4}$

This is another example. $\left(\dfrac{81}{4}\right)^{1/4}\left(\dfrac{1}4\right)^{-3/4}$ Multiply on both sides equals $\dfrac{81^{1/4}}{4^{1/4}}\cdot \dfrac{1^{-3/4}}{4^{-3/4}}$ This should be ...
0
votes
1answer
18 views

Counting with potency and simplifing

So I have the question: Simplify $(6^{n+4}) / 2^{n+5} \cdot 3^{n+2}$ I tried to write the expresion as $6^{n+4-(2n+7)}/6$, but that is wrong. So I guess I should factor it out. Perhaps $2^{2} + ...
2
votes
1answer
36 views

Why does this sequence of operations give $x^{\frac{1}{x-1}}$?

I found (purely from experimentation) that if you start with a random number and successively: Exponentiate, Raise to the power of $x$, Take the log with the same base as step one, Take the $x$-th ...
1
vote
3answers
82 views

Solve $e^{k_1/x}+e^{k_2/x}+\cdots+e^{k_N/x}=1$ for $x$

How can I solve $e^{k_1/x}+e^{k_2/x}+\cdots+e^{k_N/x}=1$ for $x$, where $N\geq 1, k_1,\ldots,k_N \in \mathbb{R}, k_1,\ldots,k_N < 0, x\in \mathbb{R}$ and $x >0$. I looked at the basic rules of ...
0
votes
1answer
45 views

Formula for reciprocal of a factorial

I was looking at some code here - https://www.codechef.com/viewsolution/6075682 when I came across this statement to calculate reciprocal of a factorial- ...
4
votes
1answer
84 views

How to prove $a^{ka}+b^{kb} \geq a^{kb}+b^{ka}$

Let $0 \leq a \leq b, 0 \leq k \leq e$ $$ a^{ka}+b^{kb} \geq a^{kb}+b^{ka} $$ It's relatively easy to prove when $b \geq 1$(every non-negative $k$ satisfies this inequality), I can't prove the other ...
5
votes
1answer
87 views

Solve $a^x+b^x=c$ for $x$

I need to solve an equation of the form $$a^x+b^x=c$$ with $a,b\in (0,1)$ and $c\in(0,2)$ (and I'm solving for $x\in\mathbb{R}_{>0}$). I know this admits a solution (details below), but it's such ...
1
vote
0answers
17 views

Sum of Bell Polynomials of the Second Kind

A problem of interest that has come up for me recently is solving the following $$\frac{d^{n}}{dt^{n}}e^{g(t)}$$ There is a formula for a general $n$-th order derivative of a composition as shown ...
3
votes
2answers
58 views

changing the power of 2 to the power of 3

this is a really simple question, I'm solving a time complexity program, to find the order of the program, however when it gets down to simplifying the mathematical part, I get stuck. I want to get ...
0
votes
1answer
35 views

Linear Recurrence Using matrix exponentiation

Matrix Exponentiation can be used to solve Linear Recurrence . I know how to solve linear recurrences like : $f(n) = f(n-k_1) + f(n-k_2) + ... +\ constant$ But i couldn't find any information on how ...
0
votes
2answers
21 views

Modular arithmetic exponentiation

Does modulus apply to exponents as well. eg Let $ xy \equiv 1 (mod\;m).$ then does $a^{xy} \equiv a^{1} (mod\;m)$ ?
0
votes
4answers
34 views

How can I calculate these large exponents with mods?

Is there a fast technique that I can use that is similar in each case to calculate the following: $$(1100)^{1357} \mod{2623} = 1519$$ $$(1819)^{1357} \mod{2623} = 2124$$ $$(0200)^{1357} \mod{2623} ...
6
votes
5answers
491 views

Solve the equation. e and natural logs

$$e^x − 6e^{-x} − 1 = 0$$ No idea how to solve this. If someone could show me the first one or two steps to push me in the right direction that would be great.
2
votes
1answer
60 views

How could I solve $x^{t-1}e^{-x} = a$ for $x$?

Consider this equation: $$x^{t-1}e^{-x} = a$$ I am aware that this is what you integrate from $0$ to $\infty$ in respect to $x$ to get the Gamma Function, but I do not want to worry about it here. I ...
0
votes
3answers
50 views

Solving $8^{2x}-2\cdot8^x+1=0$

$8^{2x}-2\cdot8^x+1=0$, I tried a lot of ways to solve this equation, like changing $8$ to $2^3$, or writing $2*8^x$ as $2*2^{3x}$ and then $2^{3x+1}$, but i'm not getting anywhere, i have the ...
0
votes
1answer
58 views

Calculate $2^n \pmod{14^8}$ with large numbers quickly

Is there a way to calculate $2^n \pmod{14^8}$ faster than binary exponentiation? The $n$ values in question are very large, for example $2^{65536}$, and the calculations have to be done around $14^8$ ...
0
votes
1answer
18 views

Indicator function - exponential form of Bernouilli [closed]

i am struggling to understand the derivation per below, which aims to explain how $Ber(x|\mu$) can be written as exp[$\phi$(x)$^t$$\theta$]. i would have interpreted that the indicator function ...
1
vote
1answer
38 views

A basic question about exponentiation

This is a silly question but under what conditions is $a^{xy}=(a^x)^y$ true, given all are complex numbers?
12
votes
2answers
170 views

Solving $z^z=z$ in Complex Numbers

I wanted to find all complex numbers $z\neq0$ such that $z^z=z$. I observed that $z=\pm1$ satisfies the equation. But I had problems when tried to find all the possible solutions since $z^z$ may take ...
-2
votes
0answers
48 views

What is $0^0$ equal to? [duplicate]

I do not think that it would make sense for $0^0$ to equal $1$, but I am not sure. I mean, at least, $\forall n \neq 0$, I believe this is true.
0
votes
4answers
59 views

Is $a^b$ larger than $b^a$ if $a<b$ and $a,b > 1$?

Is $a^b$ larger than $b^a$ if $a<b$ and $a,b > 1$? I tried this out for a few numbers and this seems to be the case. If this is true, could you show me a proof? I would be very interested. If ...
1
vote
1answer
27 views

E Scientific Exponential Notation

Gday, I have a question regarding scientific notation. Today I learnt that $a\operatorname{\mathbf{E}}b$ is the same as $a\cdot10^b$ and since myself and examiners (I'm in year 12) like neat working ...
1
vote
1answer
17 views

Does there exist any non-trivial square matrices of dimension $n$ with power cycles of less than $n$

Earlier I was faced with the matrix: $$A=\begin{bmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix}$$ Which cycles ...
1
vote
2answers
27 views

About $(x^3 - 4)^2 - x^6 + 2x^5 = 2x^5 -8x^3 + 16$

Studying polynomials I got the follows: $$ (x^3 - 4)^2 - x^6 + 2x^5 = 2x^5 -8x^3 + 16 $$ I can't understand from where we got this $-8x^3$. I got to simplify this polynomial just to: $$ 2x^5 + 16 ...
0
votes
1answer
22 views

Trailing zeros in indice question

The expression $15^{80}$ x $28^{60}$ x $55^{70}$ gives a number that ends in a string of zeros. How many consecutive zeros are in that final string? I've done this type of question with factorials, ...
0
votes
1answer
37 views

What is $R^0$ when $R=0$? [duplicate]

We say that for a number $R$, $R^0 =1$, but if $R=0$ how can $R^0$ be $1$?
3
votes
1answer
49 views

Easy difference of exponents ($a^b$ - $c^d$) for arbitrarily large numbers

I am wondering if there is an easy way to calculate the difference of two exponents, with different bases, without calculating the number. If I have $a^b$ - $c^d$, where $c^{d+1} \gt a^b \ge c^d$ ...
0
votes
2answers
27 views

Comparing large exponents

Without calculator, I have to determine which of the following is larger: $2^{350}$ or $5^{150}$ I know only very basic exponential laws, and haven't covered logarithms yet. Tried various algebraic ...
1
vote
3answers
79 views

Prove that $n^a < a^n$ for $a>1$ and $n$ big enough

How can I solve this? I'm trying to prove using limits but it's not working.. Thanks
4
votes
4answers
77 views

What is the solution to the equation $9^x - 6^x - 2\cdot 4^x = 0 $?

I want to solve: $$9^x - 6^x - 2\cdot 4^x = 0 $$ I was able to get to the equation below by substituting $a$ for $3^x$ and $b$ for $2^x$: $$ a^2 - ab - 2b^2 = 0 $$ And then I tried \begin{align}x ...
0
votes
0answers
16 views

Check whether a number could expressed as power of another two numbers [duplicate]

I found in many places how to find whther a number could be expressed as power of 2. What I need to know is, if a number is given whther that number could be expressed as a number raised to another. ...
2
votes
2answers
53 views

Linear Recurrence In Faster Time

I am trying to solve this linear recurrence using matrix exponentiation:- $$f(n) = 2f(n-1) - f(n-2) + c,$$ where $c$ is a constant. What I have come up with is this - Let the matrix $M$ be $$ ...
2
votes
2answers
28 views

Need assistance in solving exponential equation: $\frac{27^x}{9^{2x-1}}=3^{x+4}$

Find value of x: $$\frac{27^x}{9^{2x-1}}=3^{x+4}$$ My steps: $$\frac{(3^3)^x}{(3^2)^{2x-1}}=3^{x+4}$$ $$\frac{3x}{4x-2}=x+4$$ Please help me finish solving, and correct me if what I did so ...
-1
votes
1answer
30 views

Need assistance solving exponential equation: $64=0.8^d$x$100$

Solve the exponential equation: $64=0.8^d$x$100$ I tried doing: $64/100=80/100^d$ but since there is no common factor which gives these numbers with different powers I failed to find the value of ...
3
votes
1answer
47 views

Set Notation with exponent

I am looking at the function: $$f: \{5\}^2 \to \{5\}$$ it is certainly nothing too exceptional , but I find it difficult to understand what $\{5\}^2$ as a set notation and from then the whole ...
1
vote
1answer
28 views

Error in proof: Distribution of exponents for negative number [duplicate]

Here are steps of the "proof": $1=1$ $\Rightarrow 1=\sqrt{1}$ $\Rightarrow 1=\sqrt{-1\times-1}$ $\Rightarrow 1=\sqrt{-1}\times\sqrt{-1}$ $\Rightarrow 1=i\times i$ $\Rightarrow 1=-1$ At which ...
0
votes
1answer
7 views

Evaluate and simplify multiplication of exponents with base e; polar forms

$$2e^{(i×\pi/4)}×3e^{(i×\pi/6)}$$ How would I evaluate and simplify the above, and then express it in polar form? I understand $re^{i\theta} = r(\cos\theta+i\,\sin\theta)$. The question is to find ...
0
votes
2answers
80 views

How do I evaluate this:$\sum_{n=1}^{\infty}\frac{1}{n²}(e^x −1 −\frac{x}{1!} −\frac{x²}{2!}−\cdots\frac{x^n}{n!})$?

How do i evaluate this sum :$$\sum_{n=1}^{\infty}\frac{1}{n²}(e^x −1 −\frac{x}{1!} −\frac{x²}{2!}−\cdots\frac{x^n}{n!})$$ Note: I 'd surprised if it is convergent Thank you for any help.
9
votes
6answers
517 views

How to prove that $7^{31} > 8^{29}$

How can I prove that $7^{31}$ is bigger than $8^{29}$? I tried to write exponents as multiplication, $2\cdot 15 + 1$, and $2\cdot 14+1$, then to write this inequality as $7^{2\cdot 15}\cdot 7 > ...
1
vote
1answer
43 views

Why is this true: $1- (1-1/n)^{\varepsilon n} \leq \varepsilon + \mathcal{O}(\varepsilon^2)$

In my lecture notes, the following is written: $$1- (1-1/n)^{\varepsilon n} \leq \varepsilon + \mathcal{O}(\varepsilon^2)$$ as $\varepsilon \rightarrow 0$ and $n$ some fixed constant (non-negative ...
0
votes
0answers
26 views

Properties of exponentiation proof

I'm trying to prove the following: "Let $x, y$ be non-zero rational numbers, and let $n,m$ be integers. Then we have $x^n x^m = x^{n+m}$." I've managed to prove by induction the case $n,m \geq 0$ ...
0
votes
1answer
47 views

How can I raise a Taylor Series to a power?

I have recently been undertaking the challenge of finding the antiderivative of $x^x$. In doing so, I have come across the idea of raising a Taylor series to a variable exponent. I came to the ...
8
votes
5answers
1k views

Taking the square root of an imaginary number

We know that when we take the square root of a negative real number, it's realness "splits open" and an "imaginary" dimension is introduced (characterized by the presence of iota). The question is, ...
0
votes
1answer
33 views

Formula for $\sum_{i = 1}^n k^n$ [duplicate]

I know from my calculator the answer is $\sum_{i = 1}^n k^n$ = $\frac{k^{n+1}-k}{k - 1}$. I'd just like help understanding why.
9
votes
1answer
150 views

For which complex $a,\,b,\,c$ does $(a^b)^c=a^{bc}$ hold?

Wolfram Mathematica simplifies $(a^b)^c$ to $a^{bc}$ only for positive real $a, b$ and $c$. See W|A output. I've previously been struggling to understand why does $\dfrac{\log(a^b)}{\log(a)}=b$ and ...
0
votes
2answers
56 views

What determines what base the right side of this base coversion will be?

Referring to this example of positional notation on Wikipedia: There are several examples $$465\;\;\text{(base 10)} = 465\;\;\text{(base 10)}$$ But then $$465\;\;\text{(base 7)} = ...
2
votes
3answers
82 views

Sum of super exponentiation

$f(x,n)=x^{2^{1}}+x^{2^{2}}+x^{2^{3}}+...+x^{2^{n}}$ Example: $f(2,10)$ mod $1000000007$ = $180974681$ Calculate $\sum_{x=2}^{10^{7}} f(x,10^{18})$ mod $1000000007$. We know that $a^{b^{c}}$ mod ...