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Questions tagged [exponentiation]

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

2
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4answers
28 views

Approximate solution : factorial and exponentials

If z= $\dbinom{200}{100}/(4^{100})$, what is the value of z? The options are: a. $z<1/3$ b. $1/3<z<1/2$ c. $1/2<z<2/3$ d. $2/3<z<1$ How should I go about solving this type ...
1
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2answers
70 views

If $ x(9^{\sqrt{x^{2}-3}} + 3^{\sqrt{x^{2}-3}}) = (3^{2\sqrt{x^{2}-3}+1} - 3^{\sqrt{x^{2}-3} + 1} - 18) \sqrt{x}+ 6x $, what is the maximum $x$?

If $$ x(9^{\sqrt{x^{2}-3}} + 3^{\sqrt{x^{2}-3}}) = (3^{2\sqrt{x^{2}-3}+1} - 3^{\sqrt{x^{2}-3} + 1} - 18) \sqrt{x}+ 6x $$ Whati is the maximum value $x$ that fits in the equation? Attempt: $$ \sqrt{x}...
1
vote
2answers
48 views

Unintuitive behavior of x^y for x<0

I take advantage of mathematics a lot while programming, but in spite of that I'm very mathematically ignorant. Sorry if I'm asking something stupid. Often I need to exponentially amplify some values....
1
vote
1answer
18 views

Expected value of a lognormal random variable formula

Given $Z(t) \sim \mathcal{N}(0,\,1)$, why does the formula for the expected value of a lognormal random variable, $E(e^{X}) = e^{\mu + \sigma^2/2}$ give the following: $$E(e^{\sigma \sqrt{t} Z}) = e^{...
0
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1answer
42 views

A question on Greek letters used for math

The Greek letter sigma is used for repeated addition. The Greek letter pi( uppercase, not lowercase), is used to denote repeated multiplication. In the same way as the last two, which symbol is used ...
0
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3answers
44 views

How to prove the double sum of combinations is $3^n$

I have a double sum of combinations as follow $$S = \sum_{i=0}^{n}\sum_{k=i}^{n}{n \choose k}{k \choose i}.$$ I guessed and tested that $S = 3^n$, but I have no idea how to prove this. Any help is ...
5
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5answers
72 views

Finding an inverse function (sum of non-integer powers)

I have a function: $$f(x)=x^{2.2} + (1-x)^{2.2}$$ It is defined on the interval $[0,1]$. Minimum: $x=0.5, y=2*0.5^{2.2} = 2^{-1.2}$. I want to find an inverse for it. Since the function has two "...
1
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1answer
37 views

What is the value of $e^{3i \pi /2}$? [duplicate]

When solving for the value, we know that $e^{\pi i}=-1$ . I am confused as to what is the right answer when you evaluate this.I am getting two possible answers: $e^{3\pi i/2}$ = $(e^{\pi i})^{3/2}$ so ...
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2answers
22 views

Indices Question & Equation [closed]

Following is the equation $$x^{x\sqrt x}=x\sqrt{x}$$ We need to find $x$. Please help.
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0answers
62 views

Inequality for $a+b+c=3$ $3\geq (ab)^{bc}+(bc)^{ca}+(ca)^{ab}$

It's a new problem that I find interesting : Let $a,b,c>0$ such that $a+b+c=3$ then we have : $$3\geq (ab)^{bc}+(bc)^{ca}+(ca)^{ab}$$ My try : If $ab\leq 1$ , $bc\leq 1$ , $ca\leq 1$ the ...
2
votes
1answer
66 views

Power Set and Empty Set question

I have a question regarding the set of functions resulting from a set raised to a power. I think I have part of the understanding correct, however I'm having trouble interpreting $Y^{\emptyset}$. I ...
0
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1answer
53 views

Which stage in the Neumann hierarchy do powers of the reals fit in?

To be more specific than the short title, I try to gauge the size of some "normal" function spaces as e.g. found in functional analysis against set universe sizes at certain stages. For the sake of ...
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votes
3answers
59 views

How to prove relation between $(1-x)^k$ and $(1-x^k)$?

I have been trying to prove or disprove following inequality: $(1-x)^k \geq (1-x^k)$ $\forall 0 \leq x \leq 1 $ and $k \in \mathbb{N}$. I thought about taking $\log$ on both sides and comparing but ...
0
votes
1answer
43 views

$(q^{2^{n+1}})^2$ question to understand another question

I found question, that is primary question for my problem. Can't ask my question via comment to the second answer, because have not enough reputation. In proving of $$(1+q)(1+q^2)(1+q^4)\dots(1+q^{{2}^...
0
votes
1answer
49 views

Solving an inequality involving exponentials

For a homework problem, I'm supposed to compute the integral of $f(x) = x e^{-x^2}$, which I did, and got a result of $F(x) = -\frac{1}{2} e^{-x^2} + C$. Secondly, I'm supposed to find the smallest ...
0
votes
2answers
15 views

How many consecutive zeros to the left of the decimal does $125^{14} \times48^{8}$ have?

If $125^{14}\times48^{8}$ were expressed as an integer, how many consecutive zeros would that integer have immediately to the left of the decimal point? I tried: $$125^{14} \times 48^{8} = (1.25\...
0
votes
2answers
79 views

Solving $2^{2n-3}=32(n-1)$ for $n$

Just wondering is there any way to solve this equation using some algebraic or calculus tricks but without graphs. Solve for $n$ $$2^{2n-3}=32(n-1)$$ I solved this intuitively or to be more ...
2
votes
1answer
44 views

Is $X^{E} \equiv X \;(\text{mod}\; n)$ a potential problem in the RSA cryptosystem?

This is Exercises 11 from Section 7.3 of the book "Abstract Algebra: Theory and Applications" (aata-20180801) by Thomas W. Judson. Find integers $n$, $E$, $X$ such that $$ X^{E} \equiv X \;(\text{...
0
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1answer
26 views

Why rational exponents expressed in the simplest form are defined as roots but those not in the simplest form are not?

If $b$ is a nonzero real number and $p/q$ is a positive rational number, then, if $p/q$ is expressed in lowest terms, $b^{p/q}=\sqrt[q]{b^p}$. Why if $p/q$ is not expressed in lowest terms, $b^{p/q}$ ...
2
votes
5answers
73 views

What is all N that make $2^N + 1$ divisible by 3? [duplicate]

When $N = 1$, $2^N + 1 = 3$ which is divisible by $3$. When $N = 7$, $2^N + 1 = 129$ which is also divisible by $3$. But when $N = 2$, $2^N + 1 = 5$ which is not divisible by $3$. A quick lookout ...
1
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0answers
24 views

Reducing exponent in modular arithmetic (Distributed ElGamal decryption)

I'm currently trying to make an ElGamal system decrypt in a distributed fashion, so as to never reveal the secret key in play. To that effect, the secret key is put into a polynomial as the constant ...
1
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1answer
49 views

why -1 to the power of e is undefined

I'm currently just playing with the number e, I have tried $$ (-1)^e $$ But my calculator return undefined I am a bit confused by why undefined is returned. What is the logic/theory behind it? Can ...
1
vote
4answers
56 views

2^(2^(32)) mod 11

For 1420172017 mod 60 I used a logarithm rule to solve it: 1420172017 mod 60 => lg(14) a = 20172017 In essence, I calculated 20172017 mod 60 = 37 (using Euler's Theorem). Then I used that result as a ...
1
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2answers
24 views

Calculating Modulo of powers [duplicate]

I am trying to solve 1420192019 mod 60 without using a calculator. To do this, I am using the logarithm: a = bx -> x = logba => 1420192019 mod 60 => lg(14) a = 20192019 mod 60 a = 1420192019 mod 60 ...
1
vote
1answer
43 views

How do you convert 2 to power of some number to 10 to power of some number?

I realize this was asked before, but I need more precision. For example a good approximation for $2^{256}$ = $1.2$ x $10^{77}$. I've followed various methods, but the best I can do is get $1.02$ x $...
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votes
1answer
57 views

Show $8 ^ \sqrt3\in\Bbb R$, and $(-8)^\sqrt3\not\in\Bbb R$ [closed]

Show that $8 ^ \sqrt3\in\Bbb R$, and $(-8)^\sqrt3\not\in\Bbb R$.
2
votes
4answers
36 views

How to get exponent of power if base is known?

Sorry for this type of question, but I've forgotten the math basic from middle school, maybe someone can help me out. If I know the result and base, how can I calculate exponent? $2.5 = 10^x$, how ...
0
votes
2answers
51 views

what is the solution to this equation: $1 + 3^{x/2} = 2^x$?

what is the solution to this equation: $1 + 3^{x/2} = 2^x$ ? The answer is $x = 2$. I want to know the process.
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5answers
195 views

If $A =\left[ \begin{smallmatrix} 3 & -4 \\ 1 & -1\end{smallmatrix}\right]$, then find $A^n$

If $$ A = \begin{bmatrix} 3 & -4 \\ 1 & -1 \end{bmatrix}$$ Then find $A^n$ I have tried solving it using diagonalization $PDP^{-1}$ but I am getting only one independent eigenvector i.e $$ ...
1
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1answer
37 views

How to convert any base 10 integer to a rational power of 2?

For example, through (lots of) trial and error with a calculator, I figured out that: 1.551e+25 is approximately equal to 2^83.6814 Is there an equation or algorithm for a conversion like this? ...
0
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5answers
62 views

Hard exponential equation

Is there any algebraic way of solving the following equation for $x$? $$\frac{3^x+2^x}{3^x-2^x}=7$$ Apparently there is some way of solving this and I heve tried to solve it in a conventional ...
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5answers
79 views

Is it true that $n \leq 2^{n-1}$ for all natural numbers $n > 0$? [closed]

Seems to be true but I want to make sure: $n \leq 2^{n-1}, n > 0$
0
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4answers
84 views

What is the answer to $17^{16} \pmod {10}$? Is it equal to $ 9 $ or $1$

I encounter a modular arithmetic problem. which says: Find the last Digit of $17^{16}$, by intuition the last digit of a number is the remainder of the number divided by $10$. so the statement is: $...
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3answers
125 views

Why is $49^{-\frac{1}{2}}=\frac{1}{7}$?

Why is $49^{-\frac{1}{2}}=\frac{1}{7}$? I know that $\sqrt[n]{m^p}=m^{\frac{p}{n}}$ so I figured I can state the above is $-\sqrt{49}=-7$, but that is incorrect. I can't put the negative inside the ...
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3answers
61 views

$(a^b)^c$ and $a^{(bc)}$ for complex numbers

I've just stumbled across $$\left(e^{2\pi i}\right)^{\frac{1}{2}}=\sqrt 1=1\neq -1=e^{\pi i},$$do I have some error in my thoughts there or does $(a^b)^c=a^{(bc)}$ not hold for complex numbers?
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1answer
37 views

Proof of equivalence between two methods of binary to decimal conversion.

I have two binary to decimal conversion methods and want a proof - or an intuition at least - of why they are equivalent. The first method is quite intuitive to me and seems to be more popular: $[...
7
votes
1answer
81 views

For every $b$ in the power $a^{b}$, does there exist an $a$ such that the digit sum of this power is equal to $a$?

$1^0 = 1\to 1 =1$ $x^1=x\to x=x\;\forall x$. $9^2 = 81\to 8+1=9$ $8^3=512\to 5+1+2=8$. $7^4=2401\to 2+4+0+1=7$ $46^5 = 205962976\to 2+0+5+9+6+2+9+7+6=46$ $64^6 = 68719476736\to 6+8+7+1+9+4+7+6+7+...
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2answers
26 views

How can I prove that $10=2^{a}*3^{b}*7^{c}$ has infinite solutions?

Both in a unrescrited case and with the following restriction: $a+b+c=1$
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2answers
60 views

How do I solve the for the base of an exponential modular arithmetic equation? [closed]

The question is: $$10 \equiv M^5 \mod{35}$$ How do I isolate and solve for $M$?
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2answers
74 views

multiplication of exponentials for non commuting matrices

Is there any special condition for the following statement to be true for any $n \times n$ matrices? $$ e^Ae^B=e^{A+B}=e^Be^A $$ Is it always correct to say that the exponential product equals ...
3
votes
1answer
154 views

What's the derivative of $\underbrace{x^{x^{x^{x^…}}}}_n$? Are my calculations right?

The problem: To make things easier on us so we don't have to use an underbrace, I'm going to declare: $$\Psi(x,n) := \underbrace{x^{x^{x^{x^{\dots}}}}}_n$$ And we evaluate $f$ from the topright to ...
0
votes
1answer
23 views

How can you express radicals as multiplication/addition?

How can you express radicals as multiplication/addition? Most mathematical operations clearly reduce to multiplication/addition (same thing), but how do you do that for exponentials/radicals? Thank ...
0
votes
1answer
10 views

Show that the following are equivalent (exponent laws)

Determine whether the two expressions are equivalent I was able to solve this by distributing the -2(1+x$^2$) and setting it equal to the expression x$^2$-4 and getting x$^2$=2/3 and then ...
0
votes
1answer
39 views

How to prove this theorem for exponential equations

Prove that if $a^x=a^y a∈R$ then $x=y$ as in the exponential equation $2^x=2^3$ How can I prove this theorem, if you know what I mean
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1answer
21 views

Modular exponentiation with the Carmichael function

This is something I have been thinking of using in a math competition against other players so it would be very helpful to me if it was explained. How would someone reduce a problem such as $\frac{7^{...
0
votes
2answers
60 views

How to decelerate from velocity v to stop time t over distance d?

I'd be grateful for some help with this problem I am trying to solve. Let's say that I have an object travelling at a velocity v. I want that object to come to a halt in time t AND travel exactly ...
0
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2answers
72 views

What is the difference between $\dfrac{1}{3^{-2}}$ and $3^{-2}$

Stupid question but how in the world does 1/3^-2 not be the same thing as 3^-2? The first answer I got is 9, the second one I got is 0.111... Isn't the negative power just 1/3/3? What difference does ...
2
votes
2answers
77 views

Why do we take $(1+z)^{\alpha}$ as $e^{\alpha \operatorname{Log}(1+z)}$?

Let $\alpha$ be a complex number. Show that if $(1+z)^{\alpha}$ is taken as $e^{\alpha \operatorname{Log}(1+z)}$, then for $|z|< 1$ \begin{equation*} (1+z)^{\alpha} = 1 + \frac{\alpha}{1}z + \...
2
votes
1answer
35 views

Near integers in powers of binomials with radicals

This question comes out of a mathematics calendar problem that asked for the tenths digit of the expression $(17 + \sqrt{280})^{17}$. The calendar implied the digit should be 9, but after playing ...
0
votes
2answers
30 views

Exponent Problem. Verify my solution.

Is this solution correct ? please click here to see the question & my answer Sorry for the nasty handwriting (in image). $$\sqrt{a}(2a^2-4/a)\Rightarrow a^{1/2}(2a^2-4a^{-1})\Rightarrow 2(a^{5/2}-...