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

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6
votes
1answer
136 views

$1989|n^{n^{n^{n}}} - n^{n^{n}}$ for integer $n \ge 3$

Before anyone comments, yes this is kind of a duplicate of Prove that $1989|n^{n^{n^{n}}} - n^{n^{n}}$ . The problem that I'm having I don't see the $n=5$ as a counterexample. Also if anyone wants to ...
1
vote
4answers
122 views

What is the value of $\lim_{x\to 0}x^x$?

Evaluate $$\lim_{x\to 0}x^x$$ I tried by writing $x$ in terms of exponentials: $x=e^{\ln x}$ so $x^x=e^{x\ln x}$ $\lim_{x \to 0}x \ln x=\lim_{x \to 0}(\ln x +1) =-\infty$ Thus $\lim_{x\to ...
4
votes
1answer
182 views

Prove that $1989\mid n^{n^{n^{n}}} - n^{n^{n}}$

Having difficulty in proving this: $1989\mid n^{n^{n^{n}}} - n^{n^{n}}$ for all $n \in \Bbb N$. Prime factorization of $1989$ is $3^2 \times 13 \times 17$. Please Help!
8
votes
2answers
131 views

An interesting property of binomial coefficients that I couldn't prove

So when I was trying to prove the argument in this link I've come up with something. When you extract the left term from the right term, you get the term under them. What is interesting is that as ...
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
260 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 ...
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- ...
1
vote
3answers
84 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 ...
6
votes
1answer
124 views

Solution to the functional equation $f(x^y)=f(x)^{f(y)}$

Consider the functional equation problem $$ f: \Bbb{R} \rightarrow \Bbb{R}$$ $$ f(a^b) = f(a)^{f(b)},$$ when $a,b \in \Bbb{R}, a,b \ge 0.$ So far the only solution I have is the trivial $$ f(x) ...
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 ...
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 ...
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 ...
3
votes
2answers
1k views

Exponential equations with variables on both sides

I have the following: $$8^{3x+4} = 5^{4x-2}$$ How would I solve this? I tried this: $$(3x+4)\log 8 = (4x-2)\log 5$$ but have no idea where to go from there. Thank you!
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
35 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
493 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.
160
votes
22answers
13k views

Zero to the zero power - Is $0^0=1$?

Could someone provide me with good explanation of why $0^0 = 1$? My train of thought: $x > 0$ $0^x = 0^{x-0} = 0^x/0^0$, so $0^0 = 0^x/0^x = ?$ Possible answers: $0^0 * 0^x = 1 * 0^x$, so ...
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
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
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 ...
1
vote
1answer
49 views

Quick methods to check perfect 4th, 5th, 6th powers

Are there any quick modulus methods to check if a number could be a perfect power (4, 5, 6)? Preferably binary methods. For example, a perfect fourth power has to be $0, 1 \pmod{16}$ from a square ...
12
votes
2answers
171 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 ...
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?
-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 ...
4
votes
4answers
1k views

Solving $e^x + x = 5$ for $x$ without using a numerical method?

Canadian economist Mike Moffat asks on Twitter: Math nerd Q: Is there a way to solve $e^x + x = 5$ for $x$, without using a numerical method?
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 $1$?
11
votes
3answers
296 views

On the sum of digits of $n^k$

Reading another question on the sum of the digits of $2^n$ i started wondering whether there exist a $\alpha\in\mathbb{N}$ such that for every $n>\alpha$ we have $S(2^{n+1})>S(2^n)$, where ...
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
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 ...
4
votes
4answers
79 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 ...
9
votes
6answers
521 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
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 ...