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

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9
votes
1answer
150 views

For which complex $a,\,b,\,c$ does $(a^b)^c=a^{bc}$ hold?

Wolfram Mathematica simplifies $(a^b)^c$ to $a^{bc}$ only for positive real $a, b$ and $c$. See W|A output. I've previously been struggling to understand why does $\dfrac{\log(a^b)}{\log(a)}=b$ and ...
0
votes
2answers
56 views

What determines what base the right side of this base coversion will be?

Referring to this example of positional notation on Wikipedia: There are several examples $$465\;\;\text{(base 10)} = 465\;\;\text{(base 10)}$$ But then $$465\;\;\text{(base 7)} = ...
2
votes
3answers
82 views

Sum of super exponentiation

$f(x,n)=x^{2^{1}}+x^{2^{2}}+x^{2^{3}}+...+x^{2^{n}}$ Example: $f(2,10)$ mod $1000000007$ = $180974681$ Calculate $\sum_{x=2}^{10^{7}} f(x,10^{18})$ mod $1000000007$. We know that $a^{b^{c}}$ mod ...
0
votes
1answer
23 views

Harmonic Mean Solution

The harmonic mean of two positive numbers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x, y)$ with $x < y$ is the harmonic mean ...
1
vote
1answer
115 views

How can one solve $1^x=2$?

Sure, common sense says there's no solution. But, I feel, there should be one! (If there isn't, can't we construct one?)
-5
votes
3answers
66 views

What's the value of $i^i$? [duplicate]

What's the value of $i^i$?Is it real or imaginary?[$i$ here denotes imaginary number.]
0
votes
1answer
46 views

Defining exponentiation on the integers

If one defines the integers as equivalence classes of pairs of natural numbers, there is a (canonical?) way to define addition and multiplication for the integers based on addition and multiplication ...
0
votes
3answers
64 views

Solve $3^{2x} -2 \cdot 3^{x+5} + 3^{10} = 0$ for $x$

Here's the question: Solve for $x$ in $$3^{2x} - 2 \cdot 3^{x+5} + 3^{10} = 0$$ I know that you have to factor something out, I'm just not sure what that something is. Thanks in advance
0
votes
5answers
65 views

Is there any notation for general $n$-th root $r$ such that $r^n=x$?

As we know that the notation for the $n$-th principal root is $\sqrt[n]{x}$ or $x^{1/n}$. But the principal root is not always the only possible root, e.g. for even $n$ and positive $x$ the principal ...
2
votes
2answers
45 views

What is the logic/theorem/derivation behind finding the exponent of p in n! By [n/p] + [n/p^2] + [n/p^3] + …? [duplicate]

The exponent of prime number of 3 in 100! is 48. It means 100! is divisible by $3^48$ $$E_3(100!) = \left\lfloor\frac{100}3\right\rfloor + \left\lfloor\frac{100}{3^2}\right\rfloor + ...
0
votes
0answers
35 views

Exponentiation of Pascal's Triangle(in matrix form)

I want to find a pattern in subsequent exponentiations of the pascal triangle shown in the form below Matrix P[K+1][K+1]: $$ \begin{matrix} \binom{0}{0} & 0 & 0 & 0\cdots ...
-3
votes
1answer
140 views

Which is the largest power of natural number that can be evaluated by computers? [closed]

Which is the largest power of natural number that can be evaluated by computers? For example if we take a very large power of 7: $7^{120000000000}$. Can a computer calculate this number?
0
votes
3answers
55 views

Find remainder of $\frac{17^{235}}{ 23}$

I need to find remainder of $\frac{17^{235}}{ 23}$. This is supposed to be solved using the following method: $\varphi(23) = 22$ ${17}^{235} = (({17}^{22})^{10})\cdot {17}^{15}$ ${17}^{22}\equiv 1 ...
-4
votes
2answers
79 views

What is the solution of $2^{2^{2^2}}$? [closed]

Which of the following values is same as $2^{2^{2^2}}$ ? $2^6$ $2^8$ $2^{16}$ $2^{222}$ What if it is $a^{b^c}$ ? Is it $a^{(b^c)}$ or $a^{bc}$ ?
1
vote
2answers
43 views

Help with some simpler symmetric group $S_n$ problems.

I apologize if the problems seem trivial but I have not been able to find example problems or solutions to some of these questions. Could someone please confirm my attempts are correct or not? ...
2
votes
3answers
91 views

Find all $x$ such that $2^x,2^{x^2}$ and $2^{x^3}$ form $3$ terms of an A.P.

I know that if $a,b,c$ are in Arithmetic Progression, then $2b=a+c$, but in this case, I am unable to solve for $x$. Hints are appreciated. Thanks.
6
votes
6answers
123 views

What are the last two digits of $77^{17}$?

I'm trying to solve current task referenced the following but I stuck at $\displaystyle77^{17}\equiv x\pmod{100}$. As it is described on above link it uses Binomial Theorem. But I read a lot about the ...
4
votes
4answers
153 views

How to find the value of $x$ in $x^5=32$

I understand that $2^5=32$ but how would one go about finding it without doing any guessing (what if the numbers were much greater)?
-1
votes
2answers
63 views

Evaluate $2^{-n}(2^n-2^{1+n})$

The answer is $-1$, but how does one expand and simplify this expression to find this answer (what are the steps)?
-1
votes
4answers
178 views

Is $ (((\sqrt{2})^ \sqrt{2})^ \sqrt{2})^{\cdots} $ an irrational number?

It is well known that $ \sqrt{2} $ is an irrational number. Is there someone who can show me if this number: $$ \left(\left(\left(\sqrt{2}\right)^ \sqrt{2}\right)^ \sqrt{2}\right)^{\cdots} $$ is ...
-5
votes
1answer
69 views

How to solve an exponent equation? [closed]

Can someone help me with this exponent equation. $2 ^ {2x+2} - 6 ^ x - 2 \times 3 ^{2x+2} = 0$ Any ideas how to solve it? Please I really have no idea. And I have exams. Tomorrow. And now it's like ...
-1
votes
1answer
62 views

What is the value of i^i? [duplicate]

i is an imaginary number. What is $i^i$? I tried to use euler rule but the answer is strange. For example $i = e^{\frac{1}{2}i\pi}$. Using $(a^b)^c = a^{b*c}$ we got $i^i=e^{(\frac{1}{2}i\pi)*i} ...
1
vote
1answer
37 views

What are the rules of powers of powers? [duplicate]

What would $2^{3^4}$ equate to? I can think of two rules that may apply: $a^{b^c} = a^{(b^c)}$ (Making $2^{3^4} = 2^{81}\approx2.417\cdot10^{24}$) or $a^{b^c} = (a^b)^c = a^{bc}$ (Making $2^{3^4} = ...
2
votes
1answer
27 views

How to determine efficiently if the arithmetic addition and subtraction of certain powers of N can be equal to M?

I am given a number N and another number M . I have to find out if arithmetic addition and subtraction of certain distinct powers of N can lead to formation of number M . I tried different approaches ...
8
votes
5answers
423 views

How to solve this equation for $x$?$\left(\sqrt{2-\sqrt{3}}\right)^x + \left(\sqrt{2+\sqrt{3}}\right)^x = 2$

This is probably such a beginner question (and it's not homework). I've stumbled upon this: $$\left(\sqrt{2-\sqrt{3}}\right)^x + \left(\sqrt{2+\sqrt{3}}\right)^x = 2$$ How to solve this equation for ...
1
vote
5answers
78 views

Comparing two large numbers

Can you compare two large exponential numbers, like $5^{44}$ and $4^{53}$ without taking their logs?
1
vote
2answers
34 views

Combinatorial proof for summation of powers of two

I apologise if this has been posted before, but I've been poring over this problem for days now and just can't seem to get it. I'm looking for a combinatorial proof for: $2^n - 1 = 2^0 + 2^1 + 2^2 + ...
6
votes
2answers
180 views

How to find out the greater number from $15^{1/20}$ and $20^{1/15}$?

I have two numbers $15^{\frac{1}{20}}$ & $20^{\frac{1}{15}}$. How to find out the greater number out of above two? I am in 12th grade. Thanks for help!
1
vote
2answers
236 views

How is N^2/3 equivalent to 1/(N^1/3)?

I've tried to look for similar things on StackExchange and elsewhere on the net, but can't seem to find anything, so thought I'd just ask for some help on here... Someone has kindly helped me with a ...
1
vote
2answers
40 views

Converting a cryptographic hash to a string of English words: how many words are needed? (need help with exponentials)

A particular cryptographic hash is represented as a $57$ byte string, encoded as base $64$. RWSvUZXnw9gUb70PdeSNnpSmodCyIPJEGN1wWr+6Time1eP7KiWJ5eAM I want to ...
3
votes
0answers
36 views

Mapping exponential functions in polar coordinates

I tried mapping power functions onto the polar plane (i.e. converting x,y into r and $\theta$). I was successful with power functions representing $y=ax^n$ by $$r=\sqrt[n-1]{\frac ...
-1
votes
3answers
80 views

$x^x = y$. Given $y$, find $x$. [duplicate]

Title is fairly self-explanatory. For example, for $y=27$, $x$ would be $3$. Specifically I was trying to find $x$ given $y = 10^{100}$, but I'd like to know how to solve it for any value of $y$.
6
votes
4answers
128 views

When does $(x^x)^x=x^{(x^x)}$ in Real numbers?

I have tried to solve this equation:$(x^x)^x=x^{(x^x)}$ in real numbers I got only $x=1,x=-1,x=2$ , are there others solutions ? Note: $x$ is real number . Thank you for your help .
-1
votes
2answers
47 views

If $a^{p}\cdot b^{p}= (a\cdot b)^{p}$ then why $-1^{2}\cdot 3^{2}\neq (-1\cdot 3)^{2}$

If $a^{p}\cdot b^{p}= (a\cdot b)^{p}$ then why $$-1^{2}\cdot 3^{2}\neq (-1\cdot 3)^{2}\\ -1\cdot 9\neq (-3)^{2}\\ -9\neq 9$$ I'm sorry, I don't know how to put latex code.
1
vote
1answer
22 views

Is there a way to find expansion of powers of multinomials without any coefficients?

For example, $(a + b + c)^3 = a^3 + b^3 + c^3 + 3ab^2 + 3ac^2 + 3a^2b + 3a^2c + 3bc^2 + 3b^2c + 6abc$ Knowing the value of a, b and c, is there a way to find this without the coefficients i.e. $a^3 + ...
0
votes
3answers
71 views

Why does $e^{\frac10}\neq e^{\frac1{-0}}$?

I was unable to explain why this fails? I asked to it many peers and they too can't. I faced this situation when solving a kind of integration problem. Consider $x=-x$ Then $x=0$ That is, $0=-0$ ...
3
votes
3answers
154 views

Why is $\lim\limits_{x\to\infty} e^{\ln(y)} = e^{\,\lim\limits_{x\to\infty} \ln(y)}$?

In the above limit $y = x ^{\frac 1x}$. Is the above a limit or an exponent property? Thanks in advance. Context (Last paragraph): http://tutorial.math.lamar.edu/Classes/CalcI/LHospitalsRule.aspx
1
vote
1answer
26 views

What does it mean to take the power of a transition matrix? Or multiply it by a vector?

I understand the general idea that a matrix $M$ has some cell $M_{ij}$ that denotes the number of ways we can go from state $i$ to state $j$, but what does $(M^t)_{ij}$ represent? The number of ways ...
3
votes
3answers
61 views

Matrix multiplication: What is $\mathbf A^3$ and $\mathbf A^n$?

Suppose there is matrix A. I know that A2 = A $\cdot $A But what if it is A3? Is it A $\cdot $A $\cdot$A OR A2 $\cdot$ A OR A $\cdot$ A2? So basically my question is what is An?
1
vote
4answers
91 views

Modular arithmetic , calculate $54^{2013}\pmod{280}$.

How do you calculate: $54^{2013}\pmod{280}$? I'm stuck because $\gcd(54,280)$ is not $1$. Thanks.
0
votes
2answers
34 views

Maclaurin Series of $\ln(2-e^{-x}) $

I tried to solve this by using the series for $e^{-x}$ and $\ln(1+u)$ $$e^{-x}=1-x+\frac{x^2}{2}-\frac{x^3}{6}+...\\ ...
-2
votes
3answers
55 views

Inequality with the variable in both the base and the exponent. [duplicate]

I realised that what I took as terseness of my question, actually made it look like a lazy attempt to get a homework answer. The following is the edited question, hopefully up to the standards of this ...
8
votes
8answers
1k views

Compare two powers of numbers without common divisor

Which of the numbers $2^{60}$ and $3^{43}$ is greater? There is no common divisor and it must be done without a calculator.
1
vote
0answers
19 views

Maclaurin Series of $\frac{2x}{e^{2x}-1}$ [duplicate]

Calculate the Maclaurin series of $$\frac{2x}{e^{2x}-1} $$ I've tried to calculate it but the series $\frac{1}{e^{2x}-1}$ divides by 0 when x is equal to 0
6
votes
2answers
95 views

Solutions to $(4x^2+\frac{16}3x)^{\sqrt {3-x}}=(4x^2+\frac{16}3x)^{\sqrt {2x+11}-\sqrt{x+2}}$

$$(4x^2+\frac{16}3x)^{\sqrt {3-x}}=(4x^2+\frac{16}3x)^{\sqrt {2x+11}-\sqrt{x+2}}$$ I found the solutions to be $0, -\frac32, -1, -\frac43$ I can't figure out why any of those wouldn't work, but my ...
2
votes
1answer
46 views

How do we derive the sum of $3^n$ and $2^n$

I know that $\quad\sum2^n = 2 (2^n-1)$ How can we derive this summation? And also how can we deduce the summation of $3^n$ from this ? I did observe this pattern : $$ \begin{align} n &= 1 ;\ ...
6
votes
1answer
124 views

Solution to the functional equation $f(x^y)=f(x)^{f(y)}$

Consider the functional equation problem $$ f: \Bbb{R} \rightarrow \Bbb{R}$$ $$ f(a^b) = f(a)^{f(b)},$$ when $a,b \in \Bbb{R}, a,b \ge 0.$ So far the only solution I have is the trivial $$ f(x) ...
0
votes
1answer
16 views

Values of $a$ for which the equation $100^{-\lvert x \rvert} - x^2 = a^2$ has the maximum amount of solutions

$100^{-\lvert x \rvert} - x^2 = a^2$ I don't know how to approach this problem, due to the x in the exponent. I would appreciate hints more than outright solutions :)
11
votes
4answers
748 views

why is $2.2250738585072014\text{e}{-308}$ not a number? [closed]

In programming the min value of a float is: $$2.2250738585072014\text{e}{-308}$$ but when I type this into a calculator, it says Not a Number. what I am wondering ...
0
votes
0answers
23 views

Demonstration of exponentiation with induction

How can you demonstrate that $a^0 = 1$ and that $a^{-n} = (1/a)^n$ using the principle of mathematical induction?