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

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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
0
votes
1answer
25 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
votes
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'...
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
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) ...
0
votes
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
votes
0answers
16 views

Expected Value of Geometric Distribution, with Exponential Random Variable [closed]

Given $X~Geometric(1/10), E(X) = (1-1/10)/(1/10) = 9$. What is $E(X^4)$?
-4
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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 $E(X) = \sum xp_x(x)$?
1
vote
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'...
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. $...
1
vote
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 $1$. ...
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
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?
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
193 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: $$2^x+\frac{1}{2^x} = 2$$
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{...
0
votes
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
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 $\log(1 - x)$...
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 ...
-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 ...
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.
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! , ...
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 ...
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 ...
1
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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
122 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?
0
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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 $y = a \cdot b^x$?
7
votes
4answers
215 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 ...
-1
votes
2answers
45 views

$((-a)^3)^{1/2}\ne(-a)^{3*1/2}?$

The original problem is $\sqrt{(-a)^{3}}\sqrt{(-a)}$ I attempted to solve this as the following way: $\sqrt{(-a)^{3}}\sqrt{(-a)} = (-a)^{\frac{3}{2}}(-a)^{\frac{1}{2}}=(-a)^{2}=a^{2}$ However, I ...
7
votes
2answers
288 views

Fermat's Little Theorem and Euler's Theorem

I'm having trouble understanding clever applications of Fermat's Little Theorem and its generalization, Euler's Theorem. I already understand the derivation of both, but I can't think of ways to use ...
6
votes
5answers
320 views

Solve $4 \times2^x+3^x=5^x$ without any sort of calculator

Is there a way i can solve the following equation only by using high school mathematics? $$4 \times2^x+3^x=5^x$$ I tried writing $5$ as $2+3$ but didn't get any result. After that i tried to divide ...
1
vote
5answers
85 views

When is $\sqrt{x/y^2}$ equal to $\sqrt{x}/y$?

The solution to the quadratics is given by $r = -\dfrac{b}{2a}\pm\sqrt{\dfrac{b^2-4ac}{4a^2}}$, which is shortened to $r = -\dfrac{b}{2a}\pm\dfrac{\sqrt{b^2-4ac}}{2a}$, but I'm wondering how if this ...
1
vote
4answers
75 views

Why is $-32^{\frac{1}{5}} = 2$

When you factorize $-32$, you get: $-32 = (-16) \cdot 2$ $-16 = (-8) \cdot 2$ $-8 = (-4) \cdot 2$ $-4 = (-2) \cdot 2$ $-32^{\frac{1}{5}} = -2$ The reason I am asking is because you get $-4 = -2 \...
1
vote
0answers
66 views

How does $3n+1$ change the proximity of $n$ to a power of two?

This is part of an attempt to prove Collatz's conjecture. I proved a modification of Collatz's conjecture, where instead of $3n+1$ if $n$ is odd, you do $n+1$. In Collatz's conjecture, if you get to a ...
-3
votes
2answers
40 views

What mean $x^\frac{a}{b}$ where $a,b \in\mathbb{R}$? [closed]

What mean $x^\frac{a}{b}$ where $a,b \in\mathbb{R}$? a) $\sqrt[b] {x^a}$ b) ($\sqrt[b] x)^a$ Thanks to all.
2
votes
4answers
180 views

“Exponential Madness” (Gauss's challenge)

From Euler's identity, we see that $e^{i\pi}=-1$ $\Rightarrow e^{2ik\pi}=1$ [squaring both sides]. This equation surely holds for all integers $k$. EDIT: From the second equation we get $e^{1+...
1
vote
2answers
25 views

Is there an imaginary Exponent with $a^x=-b$ while a,b>0

Is there an imaginary Exponent $x$ with $a^x=-b$ while a,b>0 ? An where could something like this be used?
0
votes
2answers
54 views

$e^{e^{10^{10^{2.8}}}}$ changing $e$ with $10$

From Numberphile $$e^{e^{10^{10^{2.8}}}}$$ changing $e$ with $10$, is there a way to change only the top most number while keeping all other numbers 10? i.e what is x in : $$e^{e^{10^{10^{2.8}}}} = ...
0
votes
1answer
35 views

Calculating the last digit of exponent

I need to calculate the last digit of $723^n$.(For every positive integer $n$). If it was to calculate the last digit of $a^b$ when I know the value of $a$ and $b$,then it was easy- for example,If I ...
9
votes
5answers
291 views

Which number is greater, $11^{11}$ or $9^{12}$?

Which number is greater than $11^{11}$ or $9^{12}$? My work so far: $11^{11}=285311670611>9^{12}=282429536481$. But to verify the validity of equality should be in the range of easily ...
2
votes
4answers
53 views

Derivative of exponent

Looking to solve : $$ \frac{d}{dx}[2^{0.5x}]$$ The multiplication and X value in the exponent is confusing me. Help? Thanks!
0
votes
0answers
39 views

question concerning tetration to infinity.

So I read on-line that \begin{equation} x^{x^{x^{x^{x^{x}}}}}=2 \end{equation} where number of x goes to infinity can be solved by solving \begin{equation} x^{2}=2 \end{equation} so by the same logic \...
4
votes
1answer
125 views

Meaning of imaginary part of $\int_0^6 e^{x^3} dx$

My question is the title itself. How can it be possible that integral of real numbers can have an imaginary part?