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

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2
votes
2answers
319 views

Solving Induction $\prod\limits_{i=1}^{n-1}\left(1+\frac{1}{i}\right)^{i} = \frac{n^{n}}{n!}$

I try to solve this by induction: $$ \prod_{i=1}^{n-1}\left(1+\frac{1}{i} \right)^{i} = \frac{n^{n}}{n!} $$ This leads me to: $$ \prod_{i=1}^{n+1-1}\left(1+\frac{1}{i}\right)^{i} = \frac{(n+1)^{n+1}}...
3
votes
5answers
8k views

Calculating power without using a calculator, for example $1.05^{10}$

How to find (or estimate) $1.05^{10}$ without using a calculator? Do we have any fast algorithm for cases where base is slightly more than one? Say up to $1.1$ with tick $0.05$.
0
votes
4answers
63 views

How to differentiate x^(1/x)?

How to differentiate the following? $$x^{\frac{1}{x}}$$ (I know the answer is $\frac{1-\ln(x)}{x^{2-\frac{1}{x}}}$, but I do not understand how to get there) Attempt at solution I believe the ...
1
vote
1answer
23 views

How do I convert the following relation into a recurrence relation?

I am trying to analyse the time complexity of the fast exponentiation method, which is given as $$x^n= \begin{cases} x^\frac{n}{2}.x^\frac{n}{2} &\text{if n is even}\newline x.x^{n-1} &...
0
votes
2answers
54 views

How to tell if a function has a cusp without a graph?

For my calculus exam, I need to be able to identify if a function is indifferentiable at any point without a graph. I thought this would be rather simple, but I messed up on the question x^(2/3) ...
5
votes
4answers
135 views

Prove that $10^{340} < \dfrac{5^{496}}{1985}$

Prove that $10^{340} < \dfrac{5^{496}}{1985}$. I said since $2^{13} < 10^{4}$, we see that $5 = \dfrac{10}{2} > 10^{\frac{9}{13}}$ and so $10^{340} < \dfrac{10^{343.38}}{1985} <\dfrac{...
1
vote
3answers
102 views

Why does $a\cdot r^{-1}$ equate to $\frac {a}{r} = 1$?

Why is $a\cdot r^{-1}=1$ equivalent to $\frac {a}{r} = 1$? I am trying to write exponential functions from graphs; two points were given: $(-1,1)$ & $(-2,5)$. I am trying to find an equation ...
0
votes
0answers
16 views
-4
votes
0answers
38 views

What is E(X^a)? [closed]

In terms of expected value, is there a formula for $E(X^a)$, such that a is any real number? If not, how does one do so knowing the distribution of X, using the formula $E(X) = \sum xp_x(x)$?
2
votes
2answers
115 views

Using the basic laws of exponent

I have some problems with this question. Please help me. Thanks Simplify given expression$$ a^2 (abc)^{-2} a^3 b^7 $$ What are exponents of $a$, $b$, and $c$? I get $3,5,-2$ as exponents of $a,...
1
vote
1answer
65 views

Deep Roots…:)

Why is it that powers with very small fractional or decimal exponents all tend to one? That is, for $x \ll 1$, $a^x \approx 1$, seemingly. True, or untrue? Can anyone offer more explanation? Thanks ...
1
vote
1answer
48 views

Is this a pure imaginary number?

I've met this formula and I need to demonstrate that it is purely imaginary (it has no real part). $\frac{1}{2}\log(-\exp(i2\pi q))$, //for a real "input" q. As I don't know much about maths, what I'...
0
votes
3answers
28 views

How to divide exponents with different base numbers

Could not find a calculator online that could handle my large number. Could some help me with the solution for this very large number, I've forgotten how to divide exponentials with different bases. $...
0
votes
0answers
40 views

Real exponents on negative numbers

A textbook asks For which $p > 0$ is the solution of the IVP $$ \dot{x} = x^p, \quad x(0) = 1$$ unique and defined for all $t \leq 0$? This is simple for $p \geq 1$, but otherwise we will ...
1
vote
1answer
45 views

Simpler way to compress this exponentiation?

I am trying to find when the following is true: Let $H =(10k)^b \bmod 6(p-1)$ Let $J = 10^{H} \bmod 9p$ For some prime $p > 5$ and large $k,b$. I am trying to find when $J$ is equal to $1$. ...
-1
votes
0answers
14 views

Raising to power as producing curvature

Is there some notion or theory that deals with the connection between the exponents of a polynomial and its curvature, i.e. how much it deviates from a straight line?
12
votes
2answers
278 views

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

Before anyone comments, yes this is kind of a duplicate of Prove that $1989\mid n^{n^{n^{n}}} - n^{n^{n}}$ . The problem that I'm having I don't see the $n=5$ as a counterexample. Also if anyone wants ...
39
votes
3answers
2k views
0
votes
1answer
67 views

Last digits of n^(n-1)^(n-2)^(n-3) and so on.

If given $n$, how would I get the last digits of $n^{{n-1}^{{n-2}^{\dots}}}$, for example $$5^{\displaystyle4^{3^2}}.$$ As far as I've gotten is that the last digits tend to repeat after a while, but ...
5
votes
1answer
69 views

Proving that $2^{2a+1}+2^a+1$ is not a perfect square given $a\ge5$

I am attempting to solve the following problem: Prove that $2^{2a+1}+2^a+1$ is not a perfect square for every integer $a\ge5$. I found that the expression is a perfect square for $a=0$ and $4$. ...
0
votes
1answer
44 views

What is (a+b)^x called?

I'm wondering what (a+b)^x is called. I will need this information to study for a test, but I could not manage to Google this one out. Specifically, I need to know (a+b)^2 and (a+b)^3 are. While we'...
1
vote
2answers
46 views

Is it possible to calculate imaginary exponents without trig functions?

If I have a problem such as $2^i$, I would use the rules: $$ e^{ix} = \cos{x} + i\sin{x} \\ b^n = e^{n\ln{b}} $$ Applying this to the example $2^i$, I would let $x=\ln{2}$: $$ e^{i\ln{2}} = \cos{(\...
1
vote
6answers
186 views

Solve $2^x+2^{-x} = 2$

Need to solve: $$2^x+2^{-x} = 2$$ I can't use substitution in this case. Which is the best approach? Event in this form I do not have any clue: $$2^x+\frac{1}{2^x} = 2$$
0
votes
1answer
19 views

Calculate, simplify and expand exponents with complex numbers

Can we somehow calculate $a^z$ where z is a complex number ? Does normal exponent rules like : $$a^b\cdot a^c=a^{b+c}$$ Still work when complex numbers are in the exponent ? For example, do these ...
-1
votes
1answer
76 views

What operations have represented 2^2? [closed]

I knew = 2.2 and 2.2= 2+2 What operations have represented 2^2? ? = 3+3+3=9 ; 3.3.3=81 and continue is . what is representation for operations ? I assume to have a operation, it is called f. I ...
1
vote
4answers
78 views

Why don't parentheses matter in this case of multipication

Very basic question but can't seem to wrap my head around why this happens. Normally parentheses indicate that the operation inside must be carried out first. In this case: (a * a * a)*(a * a * a * ...
0
votes
1answer
21 views

Exponents using non integers

Let’s imagine that the number of rabbits in a field doubles each month. If we start with 6 rabbits, how many rabbits would be in the field after 10 weeks, given that 4 weeks = 1 month? I would think ...
1
vote
1answer
110 views

How to evaluate $\log(1 - x)$ in terms of $\log(x)$?

I can do this using the following relation: $$\log(1 - x) = \log(1 - \exp(y))$$ Here $y = \log(x)$ is always a negative number. However, I was wondering whether it's possible to compute $\log(1 - x)$...
8
votes
5answers
2k 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, ...
6
votes
1answer
245 views

Finding every triplet $(n,a,b)$ such that $n!=2^a-2^b$

Question : Let $n,a,b$ be positive integers. Are there infinitely many triplets $(n,a,b)$ which satisfy the following equality?$$n!=2^a-2^b$$ If Yes, then how can we prove that? If No, then how can ...
11
votes
4answers
2k views
1
vote
2answers
34 views

Exponentiation with negative base and properties

I was working on some exponentiation, mostly with rational bases and exponents. And I stuck with something looks so simple: $(-2)^{\frac{1}{2}}$ I know this must be $\sqrt{-2}$, therfore must be ...
0
votes
2answers
29 views

Find the $n$-th power of complex number

Let $z=1+2i$ be a complex number. Prove that for any $n \in \mathbb{N}^*$, the number $z^n$ has the following form: $a_n+ib_n$, with $a_n,b_n \in \mathbb{Z}$. I guess the solution lies in the ...
3
votes
1answer
91 views

Jordan form of a power of Jordan block?

How, in general, does one find the Jordan form of a power of a Jordan block? Because of the comments on this question I think there is a simple answer.
2
votes
3answers
137 views

How is Faulhaber's formula derived?

I have been wanting to understand how to find the sum of this series. $$1^p + 2^p + 3^p +{\dots} + n^p$$ I am familiar with Gauss' diagonalised adding trick for the sum of the first $n$ natural ...
3
votes
1answer
402 views

matrix exponential limit

I'm having litlle trouble here to prove the following statement: "Let $A$ an $n\times n$ matrix (real or complex). Prove that $$\lim_{n \to \infty} \left(I + \frac{A}{n}\right)^{n} = e^{A}.$$ Now I'...
0
votes
3answers
352 views

Why is X raised to the power of 0 = 1?

So as the topic states, why is 5^0 = 1 and not 5 or 0? Is it only because the other exponential laws wouldn't work if it was not the case?
5
votes
2answers
361 views

Powers in ASCII text

I have a problem reading a discussion forum post. Namely, in the ASCII text, is 2^3^4 the same as $(2^3)^4$ or $2^{3^4}$?
54
votes
10answers
3k views

What Is Exponentiation?

Is there an intuitive definition of exponentiation? In elementary school, we learned that $$ a^b = a \cdot a \cdot a \cdot a \cdots (b\ \textrm{ times}) $$ where $b$ is an integer. Then later on ...
1
vote
2answers
35 views

How to solve $1/{4a^{-2}}$

Can i write $1/4a^{-2}$ as $4a^2$ ? Or is the right answer to do it like: $$1/4a^{-2} = 1/4 \cdot 1/a^{-2} = 1/4 \cdot a^2 = a^2/4$$ In the problem there is no parenthesis around $4a$ but assuming ...
2
votes
5answers
138 views

What is the actual meaning of power which is less than 1? [duplicate]

$3^4 = 3 \cdot 3 \cdot 3 \cdot 3$ $3^2 = 3 \cdot 3$ $3^1 = 3$ But my problem is for below 1: Example: $3^{0.7} = \ ?$ Can you please explain it to clear my doubt?
0
votes
1answer
23 views

How to simplify modular exponential expressions with factorial as exponents?

Said I have the following expression: n = 39^50! mod 2251 By Fermat's Little Theorem: 39^2250 = 1 mod 2251 Solving: 2250 = 50.45 n = 39^50.49.48.47.46.45.44! mod 2251 Let b = 49.48.47.46.44! , ...
1
vote
1answer
55 views

Function $f: \mathbb{Z} \to \mathbb{Z}^n$ related to $\sum_{k=1}^{x} k^n$.

The sequence $\{a_0,a_1,...a_x\}$ has closed form $a_n=\sum_{i=0}^{\infty} \Delta^i(0) {n \choose i}$ where $\Delta a_n$ denotes the operation mapping $a_n$ to $a_{n+1}-a_n$ and $\Delta^i(0)$ is ...
0
votes
2answers
61 views

Solve exponential inequality

I've come across the following exponential inequality and, unfortunately, I encountered some difficulties trying to solve it. $$ \left | x \right |^{2x^2 - 3x + 1} \leq 1, x \in \mathbb{R} $$ I ...
24
votes
12answers
3k views

What is the accepted syntax for a negative number with an exponent?

A friend is taking a college algebra class and they are teaching him that $$-3^2 = -9$$ Their explanation is: $$-3^2 = -(3^2) = -9.$$ It has been a long time for me but I thought that in the ...
1
vote
5answers
45 views

Simplify the following expression

$$\left(\frac{4}{27}\right)^{3/2}$$ I am trying to figure out how to solve this problem and my teacher was not explaining it well enough for me to grasp it. Can anybody maybe help me with step-by-...
2
votes
1answer
121 views

Where does this equation come from? [duplicate]

Since I study 3 years i ask myself very often where does this equation come from? $$e^{i\theta} = \cos(\theta)+i \sin(\theta)$$ Is it found by series expansion?
1
vote
2answers
2k views

What better way to check if a number is a perfect power?

What better way to check if a number is a perfect power? Need to write an algorithm to check if $ n = a^b $ to $ b > 1 $. There is a mathematical formula or function to calculate this? I do not ...
7
votes
4answers
214 views

Find the value of $x$ if $x^{x^4} =4$

Find the value of $x$ if $x^{x^4}=4$. Given options are $2^{1/2}$ $-2^{1/2} $ Both 1. & 2 None of the above From option verification, we get option 3. as correct one. But is there any real ...
0
votes
2answers
37 views

Is it possible to convert $y = a^x + b^x$ to the form of $y = a \cdot b^x$?

I don't think this is possible, but I wanted to ask people who know more than me. Is it possible to convert $y = a^x + b^x$ to the form of $y = a \cdot b^x$?