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

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0
votes
2answers
16 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
255 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
26 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
37 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
37 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
88 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 $14^8$ ...
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
78 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
520 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 ...
0
votes
1answer
22 views

Harmonic Mean Solution

The harmonic mean of two positive numbers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x, y)$ with $x < y$ is the harmonic mean ...