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

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4answers
48 views

number raised to a logarithmic fraction

I know and saw that $n^\frac{1}{\lg n} = 2$ but I don't understand how this can be. What am I missing?
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4answers
49 views

Help with simplification of a rational expression (with fractional powers)

Can you please help me see what I don't see yet. Here's a problem from a high school textbook (ISBN 978-5-488-02046-7 p.9, #1.029): $$ \frac{ (a^{1/m}-a^{1/n})^{2} \cdot 4a^{(m+n)/mn} }{ (a^{2/m}-a^{...
12
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0answers
128 views

Is $2^{16} = 65536$ the only power of $2$ that has no digit which is a power of $2$ in base-$10$?

I was watching this video on YouTube where it is told (at 6:26) that $2^{16} = 65536$ has no powers of $2$ in it when represented in base-$10$. Then he - I think as a joke - says "Go on, find another ...
29
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11answers
972 views

What are the Laws of Rational Exponents?

On Math SE, I've seen several questions which relate to the following. By abusing the laws of exponents for rational exponents, one can come up with any number of apparent paradoxes, in which a number ...
0
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5answers
54 views

How solve for x in an Infinite exponent

How would one solve for x in the following equation: $x^{x^{x^{x^{\cdots}}}} = 4$ The exponent continues forever... So what is the value of x? Thank you for helping
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2answers
23 views

Properties of exponents when dealing with induction.

This will most likely be a simple question for most of you. While watching my lecture today the white board cut out and the instructor didn't explain the final step in an example. He went from $(3^{2^...
0
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1answer
26 views

Adding Similar Elements

Please bear with me my maths is very rusty. If we have $x^2 + x^2$ this should be $2x^2$ meaning if powers are the same we can do the addition. The question I have is how is $2^{x+1} + 2^{x+1} = 2....
3
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1answer
38 views

How to solve the equation $xy = 1, x^{2x-y} = y^{2(x-y)}$

I have the following equation that I don't know how to solve: $$ \begin{cases} xy = 1 \\ x^{2x-y} = y^{2(x-y)} \end{cases} $$ Here's what I've tried (but my mathematical instinct tells me that I didn'...
2
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2answers
321 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
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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
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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
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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
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2answers
55 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
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4answers
136 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{...
0
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3answers
106 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
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0answers
16 views
2
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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
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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
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1answer
49 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'...
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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
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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
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1answer
48 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 $1$. ...
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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
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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
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3answers
2k views
0
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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
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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
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6answers
197 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
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1answer
20 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
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1answer
77 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
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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
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1answer
23 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
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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
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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
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4answers
2k views
1
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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
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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
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3answers
138 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
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1answer
404 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
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3answers
355 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
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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
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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
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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
139 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
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1answer
24 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
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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 ...