Questions about exponentiation

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2
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3answers
62 views

Convert from high exponent of base $10$ to base $2$.

Is there an efficient way to convert from a high exponent of base $10$, to base $2$? Both in exponent notation. Here's an example: If I have a number that's $10^5$ or even $10^{100}$, and I wanted to ...
1
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1answer
48 views

Simplify an expression.

Don't know how to do this. Simplify the expression, show steps: $$\large \dfrac {a^{-\frac 14}a^{\frac 32}}{a^{\frac 13}}$$ Write the answer using only positive exponents. Assume that all variables ...
0
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1answer
16 views

A complex exponential equation

Find x in the following equation: (1+root(3))^x + 2^(x-1)*(2+root(3))^x = 4 I have no idea what this site's "quality" standards are, which are preventing me from posting the question, so I'm typing ...
3
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2answers
58 views

Is there a proof that $n^xm^x = (n^x)^{(\log(mn)/\log(n))}$?

This isn't a homework question, just something I'm curious about, but you can treat it that way if you like. So the other day I was playing with my calculator and I noticed that $$ 2^x10^x = ...
1
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2answers
44 views

Solve $3^{1/4} \cdot 9^{-5/8}$

I don't understand how to solve $3^{1/4} \cdot 9^{-5/8}$. Help please? I have tried many different things, but they're not working. Once I plug the problem into a math equation solver, the answer ...
0
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1answer
27 views

Equations with exponents

I can't remember how to solve equations that have exponent and a variable in them. This is somewhat embarrassing, because this used to be really easy for me. I know that logarithms are involved I just ...
1
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1answer
52 views

Two variable integer equation

I have the following equation: $$ p^q(2^{q-1}-1)=9p^7q $$ I need to solve for $p$ and $q$. $p$ and $q$ are integers. I think I could take the case $p=0$ separately and for that one $q$ could be ...
3
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1answer
39 views

Are there any methods to exponentiate a real number with a number from an arbitrary field?

How can I take the following exponent, for some real-valued number a? $$a^{3+2j-9k+3i}$$ over the field of quaternions, or any field for that matter? On wikipedia we are given the following formula, ...
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2answers
42 views

Proving that if $a>1$ and $x>y$ then $a^x>a^y$

I got this assignment for homework and I can't find this anywhere around the web. Prove that if $a>1$ and $x>y$ then $a^x>a^y$. I started the assignment but I'm not sure it's enough: ...
2
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0answers
76 views

Sums of powers.

Here's the problem: Show that $19^{19}$ is not the sum of a fourth power and a positive or negative cube. I'm just not really sure how to start approaching this problem. Does anybody have any ...
3
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3answers
122 views

Proofs for $0^0 =1$? [duplicate]

Everyone knows the following: $$0^x = 0 \quad \wedge \quad x^0 = 1 , \quad\forall x \in R^*$$ One morning, I wake up asking myself the question "$\text{What is $0^0$, then?}$". So, I did what any ...
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2answers
52 views

Complicated rational exponents

How is this m/n equals to m × (1/n)? any logical proof for this? which draws this conclusion ...
0
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1answer
415 views

variable with negative exponent in the denominator moved to nominator and vice versa

The top and bottom of the fraction both contain negative exponents. Since $c^{-3}$ on the bottom has a negative exponent, it is moved to the top of the fraction (numerator). Since the $d^{-3}$ on the ...
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3answers
83 views

Easy exponents question

I have the GRE Friday... I got hung up on this easy exponents problem (I think it was these exponents, don't recall exactly) $$\frac{6^{14}}{2^7 * 3^5} = ? $$ The answer is $(2^7)(3^9)$.
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2answers
180 views

Help with limit of radical function

$$\lim_{x \to \infty} \frac{\sqrt{4x^{4}+3}}{5x^2+3}$$ $$= \lim_{x \to \infty} \frac{(4x^{4}+3)^{1/2}}{5x^2+3}$$ $$= \lim_{x \to \infty} \frac{(\frac{4x^{4}}{x^{1/2}} +\frac{3}{x^{1/2}})^{1/2} ...
0
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1answer
70 views

The shape of a graph of a function with $n$th-roots?

Not just these type of functions: $$\sqrt[3]{x}=x^{1/3} \;\;\;\text{and} \;\;\; \sqrt[8]{x}=x^{1/8}$$ But also more complicated expressions, like expressions that have $n$th roots inside of ...
1
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3answers
69 views

How to prove that $\|e^{X+Y}-e^X\| \leq \|Y\| e^{\|X\|} e^{\|Y\|}$?

A couple of questions from the Wikipedia "matrix exponential" article: In the part of the article I linked to, they mention that to conclude that every matrix in $GL(n)$ has a logarithm (though not ...
1
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0answers
23 views

Golden-Thompson inequality and Lieb's theorem

On the [Wikipedia article][1] on "matrix exponential", they draw a relation between the Golden-Thompson inequality and Lieb's theorem. My questions are: It mentions that Lieb's thoerem "accomplishes ...
0
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1answer
64 views

Exponentiation by squaring

I need to calculate $7^{2012}$ mod $13$ by hand using exponentiation by squaring, but I cant seem to figure it out. I started with this but I don't know for sure if its correct or where it's going. ...
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3answers
43 views

What is the summation of the following expression?

What's the summation of the following expression; $$\sum_{k=1}^{n+3}\left(\frac{1}{2}\right)^{k}\left(\frac{1}{4}\right)^{n-k}$$ The solution is said to $$2\left(\frac{1}{4} ...
31
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14answers
3k views

Which is larger? $20!$ or $2^{40}$?

Someone asked me this question, and it bothers the hell out of me that I can't prove either way. I've sort of come to the conclusion that 20! must be larger, because it has 36 prime factors, some of ...
3
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1answer
33 views

Is the fraction of the irrational exponentiations of two coprime integers by a rational an irrational?

Consider two strictly positive integer coprimes $n, m\in\mathbb{N^*}$ and a rational $r=\frac{p}{q}\in\mathbb{Q}$. Consider furthermore that the three number statifies the following condition: ...
2
votes
2answers
89 views

Can the exponentiation of an integer by a rational be a non-integer rational?

Consider a strictly positive integer $n\in\mathbb{N^*}$ and a rational $r=\frac{p}{q}\in\mathbb{Q}$. My question is the following: what is the nature of $n^r$? My first guess is that $n^r$ is an ...
0
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2answers
36 views

Can you simplify a expression with an exponent that is divided by a number?

As the title suggests, I have $\;a^{(b/c)}.$ Is there any way to simplify this so that there is no dividing in the exponent?
0
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2answers
131 views

How to find $x^4+y^4+z^4$ from equation?

Please help me. There are equations: $x+y+z=3, x^2+y^2+z^2=5$ and $x^3+y^3+z^3=7$. The question: what is the result of $x^4+y^4+z^4$? Ive tried to merge the equation and result in desperado. :( ...
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0answers
19 views

Rational exponentiation?

Consider the following operation: $\left(\frac{a}{b}\right)^\frac{n}{m}$ where $a, n\in\mathbb{Z}$ and $b, m\in\mathbb{N^*}$. My question is: when the result is a rational number, how (formula or ...
2
votes
2answers
280 views

How to compute $2^{\text{some huge power}}$

I have to compute $$2^{p-1} \mod p$$ and show by Fermat's little theorem that $p$ isn't prime. I know what the question is asking but I'm not sure how to reduce the exponent on $2^{p-1}$ to a more ...
2
votes
5answers
116 views

How to quickly identify perfect powers

In a test I'll take there may be a question such as the following: A perfect power is an integer that can be written as $a^b$, $a$ and $b$ being integers greater or equal to 2. One of the ...
2
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1answer
71 views

Trouble solving polynomial equation with exponent

I'm having trouble solving this equation.It looks simple, but I just can't find the answer.Can someone help me? $$9x^4-13x^2+4 = 0$$
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1answer
86 views

Is $2^n \mod m \equiv (2^{n/2} \pmod m ) ^ 2 \pmod m$?

I'm trying to write a procedure that solves (2^n - 1) mod 1000000007 for a given n. n can ...
1
vote
2answers
161 views

What does $i^i $ equal and why? [duplicate]

I've been reading up on why the value of 0^0 is controversial (see Zero to the zero power - Is $0^0=1$?) and I wondered: is it possible for $i^i$ to have a value? I plugged it into a TI-83 calculator ...
5
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2answers
33 views

Upper bound for product of exponents

From here we have the bound $$\left(1-\frac1N\right)^N\leq e^{-1}$$ where $N$ is a positive integer. Written another way, it is ...
0
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0answers
92 views

Integral involving Modified Bessel function, exponential and power

I am trying to evaluate the following integral: $$ \int_{b y}^\infty \frac{e^{-x}(-1+I_0[2\sqrt{bx}]-\sqrt{bx}I_1[2\sqrt{bx}])}{x^2}dx $$ Even though there are a lot of integrals involving the ...
0
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1answer
29 views

Question about basic exponential/logarithm properties

Solve for $k$: $$e^{k/2}=a$$ Solution: $$e^{2k}=a$$ $$ k/2 = \mathbf{ln}a$$ $$ k=2\mathbf{ln}a$$ $$= \mathbf{ln}a^2$$ My question is: why does $2\mathbf{ln}a = \mathbf{ln}a^2$? Why can you ...
0
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0answers
65 views

Development of imaginary exponent without appealing to “ambiguity” between $i$ and $-i$

Is there a way to develop the definition of the imaginary exponent, $z^i$, for complex $z$, that does not appeal to the notion that $i$ and $-i$ are "qualitatively indistinct" and that does not rely ...
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0answers
24 views

Math equations of electron scattering

I'm trying to figure out the missing step here, in a problem about X-ray crystallography. I am referring to the attached image: In the image, A= electron density, Z= distance traveled, λ= X-ray ...
0
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1answer
44 views

Unary minus on squared number

According to algebra as I know it, $-2^2 = 4$, but most calculators expand this to $-2 * 2 = -4$, which yields a different answer. This is because of the order of precedence. In traditional math, ...
3
votes
1answer
88 views

find value (-2)^-(2)^(-2)

Find the value of $(-2)^{-(2)^{(-2)}}$. Is it 16/8/-8/none? My attempt: $a^{-x}=\frac1{a^x}$, so, $(-2)^{-(2)^{(-2)}}=(-2)^{\frac{-1}{2^2}}=\frac{1}{(-2)^{\frac14}}$. That is, I would pick 'none ...
5
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4answers
74 views

Convergence of exponential matrix sum

Let $A$ be an $n\times n$ matrix. Consider the infinite sum $$B=\sum_{k=1}^\infty\frac{A^kt^k}{k!}$$ Each term $\dfrac{A^kt^k}{k!}$ is an $n\times n$ matrix. Does the sum $B$ always converge? (i.e. ...
0
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1answer
38 views

Modular Exponentiation Equivalence Problem

Find the integer $a$ such that $0 \leq a < 113$ and $102^{70} + 1 \equiv a^{37} \bmod{113}$. I started off by using modular exponentiation to realize that the left side of the congruence is ...
2
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3answers
61 views

computing $2^{170}+ 3^{63}\pmod {19}, 3^{175} + 2^{73} \pmod {17}$, etc… by hand

I came across several questions like this in the problem section of a book on coding theory & cryptography and I have no idea how to tackle them. There must be a certain trick that allows for ...
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0answers
46 views

Double summation of elementary functions

I am finding some trouble on calculating the following double summation: $ \sum_{k=1}^\infty \frac{b^k}{k!k}\sum_{n=0}^{k-1}\frac{(b*x)^n}{(n-1)!} $ Note that the inside sum gives: ...
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0answers
61 views

What is the number theory behind this?

I am given $3^{1000}$ and asked to find, in base $2$, now many digits it takes to represent this number. According to Wolfram, it is $1585$, but I don't know why. I understand that $2^n$ would be ...
0
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1answer
32 views

Equation with a division in exponent

I have an equation: 10,9 * 2^(x/1,5) = 1000 and want to calculate the value of x. x being in the exponent is my problem. How can I get to something like: x = ...
2
votes
4answers
155 views

Raising a Complex Number to a Decimal Value

So for my class i have to make a java program that deals with complex numbers. I finished getting the root and power and i was wondering how to do a method that deals with powers such as 2.56. Now im ...
0
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1answer
66 views

big-O proof with power functions

I was wondering if anyone could show a proof for why $a^x$ is $\mathcal{O}(b^x)$ if $a$ and $b$ are constants and $a < b$. In other words, with power functions, does the function with the largest ...
3
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3answers
172 views

$x^{y^z}$: is it $x^{(y^z)}$ or $(x^y)^z$?

Of the following, why is a usually considered true, and for what reason other than "tradition" and "more convenient"? a: ${x}^{y^z} = x^{(y^z)} \neq {(x^y)}^z$ b: ${x}^{y^z} = {(x^y)}^z \neq ...
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1answer
103 views

How to solve $5^n - 5^{n-3} = 5^{n-3} *124$

how is $$5^n - 5^{n-3} = 5^{n-3} *124$$ Can anybody provide a step by step solution.I will greatly appreciate if any online source for such material is provided. Regards
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2answers
89 views

x raised by the power of y equals infinity.

How many zeros are there in this equation? 10000^999 In my calculator it says infinity but that doesn't seem right.
0
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0answers
46 views

Numerical integral involving exponential of a square matrix

I am trying to evaluate an integral involving exponential of a square matrix and numerical evaluation would be fine. I am bit concerned that I may do something wrong as it involves exponential of ...