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

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1
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1answer
30 views

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

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
22 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 ...
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votes
1answer
67 views

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

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
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3answers
47 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 ...
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votes
2answers
74 views

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

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
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2answers
40 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
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3answers
90 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
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6answers
115 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
148 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)?
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2answers
56 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)?
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4answers
177 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 ...
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votes
1answer
62 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
59 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
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1answer
34 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} = ...
0
votes
1answer
127 views

What is the formula for summation of $n^n$? [closed]

How should I calculate: $$1^1+2^2+3^3+\dotsb +n^n$$ What is the formula for this submission?
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
414 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
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5answers
76 views

Comparing two large numbers

Can you compare two large exponential numbers, like $5^{44}$ and $4^{53}$ without taking their logs?
1
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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 + ...
5
votes
2answers
169 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!
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2answers
234 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
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2answers
35 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
33 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 ...
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votes
3answers
79 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
124 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 .
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2answers
46 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
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1answer
17 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
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3answers
68 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
145 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
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1answer
25 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
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3answers
59 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
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5answers
87 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.
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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}+...\\ ...
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votes
3answers
54 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 ...
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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.
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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
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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
44 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 ;\ ...
4
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0answers
78 views

Solution to following functional equation

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
735 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 ...
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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?
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2answers
21 views

exponent problem solving

I came across a problem; $a^x=b^y=c^z$ and $b^2=ac$. It is required to show $\frac{1}{x}+\frac{1}{z}=\frac{2}{y}$. I have tried the following steps- \begin{equation*} b^2=ac \\ b=\sqrt{ac} \\ ...
1
vote
1answer
44 views

Can we write “fractional root” symbol in math?

Fractional exponents are legit but I have never seen fractional roots, so I just wonder if we can write fractional roots such as this: It sometimes can be convenient to think about too.
2
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1answer
60 views

Evaluate the limit $\lim_{n\to\infty} \frac{3^n}{2^n+3^n} $

It seems reasonable to assume that $$\lim_{n\to\infty} \frac{3^n}{2^n+3^n} $$ goes to zero but I can't figure out how to prove it.
1
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1answer
38 views

How to solve for x in $2^{2x^2}+2^{x^2 + 2x + 2} =2^{5+4x}$

This is the question: $$\large{2^{2x^2}+2^{x^2 + 2x + 2} =2^{5+4x}}$$ What I did was put $~\large{2^{x^{2}}=t}$ From this, I got, roots of the quadratic: $$\large{-2^{x+1}\pm~\left( ...
0
votes
2answers
33 views

Continued addition and under rooting of 12

$\sqrt{(12 + \sqrt{12......})}$ and so on.... How do I find its answer? This is a question in our class VII mats book. P.S. - Answer is 4
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2answers
48 views

Square Root of $320$

Given, $$\sqrt{5} = 2.236$$ $$\sqrt{320} = 2^3 \times \sqrt{5} = 8 \times 2.236 = 17.888$$ This is the explanation provided in my school book. Could someone please elaborate ? Thanks in ...
0
votes
4answers
31 views

Why is 'something per hectare' denoted with a negative exponent ( $ha^{-1}$)?

Quick question.... why is it that something per hectare is shown as having a negative exponent, $ha^{-1}$? For example, on this page: http://www.ipcc.ch/ipccreports/sres/land_use/index.php?idp=12 1 ...
0
votes
3answers
56 views

Formula for exponents?

Is there a formula for exponents that works with both negative and positive powers? I have tried searching online but only found: If positive do this, if negative do this. Thanks. EDIT: Ah, I see ...