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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1answer
40 views

Pandigital exponent solution?

I don't have Reviewing C++ By Alex Maurea book. But someone in facebook post a question from this book which is following You can see the puzzle at page 542 problem 29. Now I think the answer ...
2
votes
4answers
187 views

Which is larger :: $y!$ or $x^y$, for numbers $x,y$.

This is a generalization of this question :: Which is larger? $20!$ or $2^{40}$?. No explicit general solution was presented there and I'm just curious :D Thank-you. Edit :: I want a most-general ...
2
votes
1answer
53 views

Problem finding limit - which function is asymptotically larger

I have a homework question, so please don't answer fully but I would appreciate a push in the right direction. Basically we need to figure out if $n^{n+\frac{1}{2}}e^{-n}$ is larger,smaller, or equal ...
3
votes
4answers
107 views

Calculating $\log_7 125$

So the problem asks to calculate $\log_7 125$. It's multiple choice and the options are $2.48$ $4.75$ $1.77$ $2.09$ Given that $7^2 = 49$ and $7^3 = 343$, the answer must be either option 1 or 4, ...
7
votes
6answers
238 views

What is $(-1)^{\frac{2}{3}}$?

Following from this question, I came up with another interesting question: What is $(-1)^{\frac{2}{3}}$? Wolfram alpha says it equals to some weird complex number (-0.5 +0.866... i), but when I try ...
0
votes
1answer
48 views

Fractional exponentiation in modular arithmetic

Does raising a modular expression to a fraction mean anything? For example, $a\,\,mod \,\,N$ raised to $1/b$ where $b>0$. Does this violate the rules of modularity?
6
votes
2answers
119 views

What's the first digit of 2410^2410?

The first digit means the left most digit. 2410 is just an example and it can be replaced by any other numbers. Can any one help me to solve it?
-1
votes
1answer
54 views

How do I solve for $t$ and $s$ in $y = x^{-t/s}$?

I have $$y = x^{-t/s}$$ How do I solve for $t$ and $s$ in terms of the other variables?
1
vote
2answers
81 views

Why is $2^a > a^3$?

I found this rather interesting and maybe, a bit too obvious for some people property about 2 raised to some power. $2^a > a^3$, if $a=0,a=1 \text{ or } a\ge 10$ .($a \in N$) I seem to get a bit of ...
4
votes
4answers
134 views

negative exponent problem

$$\sqrt{\frac{1}{3^0 + 3^{-1} + 3^{-2} + 3^{-3} + 3^{-4}}}$$ Does this equal = $$ \begin{align*} & \sqrt{3^0 + 3^1 + 3^2 + 3^3 + 3^4} \\ =&\sqrt{1 + 3 + 9 + 27 + 81} \\ =&\sqrt{121} \\ ...
1
vote
4answers
77 views

What exponent should I raise $26$ to in order to equal $2^{76}$?

I want to figure out how long an all-caps password needs to be to equal $2^{76}$ bits of security. I would type this into Wolfram Alpha, but I'm not sure what function to use or if it can compute ...
1
vote
3answers
62 views

Exponential Equations

I solved this , but I am not sure if I did in the right way. $$2^{2x + 1} - 2^{x + 2} + 8 = 0$$ $$2^{x + 2} - 2^{2x + 2} = 8$$ $$\log_22^{x + 2} - \log_22^{2x + 2} = \log_28$$ $$x + 2- 2x - 2 = ...
1
vote
1answer
34 views

What contributes more to a increase in value of a exponentiation: a increase of base or exponent?

Firstly, my apologies for being a programmer messing in math land. I was wondering whether the value of a exponentiation is increased more by a increase in the base or exponent (I believe this is a ...
1
vote
2answers
61 views

Polar form for complex number with variable exponent

This might be an easy question but I'm having trouble showing that $\left(1+\frac{i\theta}{m}\right)^m$ has the angle $ m\arctan\left(\frac{\theta}{m}\right)$ in polar form on the complex plane. ...
12
votes
7answers
381 views

integral of $x^2e^{-x^2}~dx$ from $-\infty$ to $+\infty$

I know that the $$\int^{+\infty}_{-\infty}e^{-x^2}~dx$$ is equal to $\sqrt\pi$ It's also very clear that $$\int^{+\infty}_{-\infty}xe^{-x^2}~dx$$ is equal to 0; However, I cannot manage to ...
7
votes
2answers
194 views

Can $2^k + 2^j$ be expressed as $2^n$?

If we are given $j,k \geq 0, j> k$ and $j,k$ are integers, can $2^j + 2^k$ be ever expressed as $2^n$ where $n \geq 0$ and is an integer? What I said: Suppose it can. Then for some $0 \leq n ...
1
vote
3answers
53 views

Calculating $\phi(x^y)$

I know how to compute $\phi(x)$ like $\phi(21)$ or $\phi(7)$ but how can I compute $\phi(x^y)$. Specifically how can I compute $\phi(5^{20})$?
3
votes
3answers
95 views

Exponentials in complex numbers

If $\displaystyle z-\frac1z=i$, then find $\displaystyle z^{2014}+\frac{1}{z^{2014}}$. The answer should be in terms of $1, -1,\;i\;or\;-i$. I am not able to understand how to simplify the given ...
0
votes
1answer
90 views

Solve some unusual log/exponential equations

I understand about log and exponential equations/functions, but I can't solve these (the numbers are just examples, of course): $ 4^x = x + 10$ $x^x = 3$ $(2x + 3x^2)^{x + 1} = (x - x^3)^{x^2}$ Are ...
8
votes
6answers
327 views

Find $x$ and $y$ in $2^{x-y} + 1 = 2^x,$ where $x,y$ are integers

I have no idea what to do now. Is there any way to find the integers $x$ and $y$ by factoring? Thank you.
1
vote
1answer
211 views

Negative Base to non-integer power

I'm looking to consistently solve the m^n case, including conditions where m is negative and n is non-integer. I'd like to, additionally, catch the error when it isn't possible. Some examples to ...
5
votes
2answers
134 views

Extending exponentiation to reals

I've been reading through a course on exponential functions, starting from integer-valued exponents to rational ones as in: $x^r$ from $r\in \Bbb{N}$ to $\Bbb{Z}$, and combining them to rigorously ...
1
vote
2answers
70 views

Recursive definition of recursively defined operations

The recursive definitions of addition, multiplication, and exponentiation usually stop after exponentiation ("${\small+}1$" to be read as "the successor of"): $x \boldsymbol{+} (y\ {\small+}1) := (x ...
1
vote
2answers
171 views

How can you prove that a value raised to $\frac{1}{n}$ is the n'th root of $x$?

I know that if you raise a value to $\frac{1}{2}$ for example, you take the square root, but that is not what I am asking, what I am asking is; what are you actually doing when raising a value to ...
2
votes
1answer
37 views

How many people are included in three-hop network analysis? [closed]

According to the ACLU, if you have 40 contacts in your network, and the NSA collects data from your network to three degrees of separation, then it could collect data on up to 2.5 million people. How ...
3
votes
4answers
216 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
vote
1answer
50 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 ...
3
votes
2answers
71 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
vote
2answers
54 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
votes
1answer
35 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
vote
1answer
55 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 ...
13
votes
1answer
402 views

If $n^c\in\mathbb N$ for every $n\in\mathbb N$, then $c$ is a non-negative integer?

Supposing that a real number $c$ is given, is the following true? "If $n^c$ is a natural number for every natural number $n$, then $c$ is a non-negative integer." Though this seems true, I can't ...
3
votes
1answer
47 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, ...
1
vote
2answers
44 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
votes
0answers
90 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 ...
2
votes
3answers
169 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 ...
0
votes
3answers
74 views

Complicated rational exponents

How is this $\frac{m}{n}$ equals to $m \times \frac{1}{n}$? Any logical proof for this? Which draws this conclusion: A fractional exponent like $\frac{m}{n}$ means do the $m^{\text{th}}$ power, ...
0
votes
1answer
2k 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 ...
1
vote
3answers
103 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 $(2^7)(3^9)$.
0
votes
2answers
943 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
votes
1answer
114 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
vote
3answers
78 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
vote
1answer
67 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 ...
1
vote
1answer
111 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. ...
1
vote
3answers
48 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} ...
39
votes
14answers
4k 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
votes
1answer
78 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
100 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
votes
2answers
42 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
votes
2answers
459 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. :( ...