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

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1answer
32 views

Exponents with base close to $1$.

I was just fiddling around with a calculator and calculating powers of numbers really close to $1$ like $1.01,1.001\dots$ trying to find at what value they exceed $2$. This got me thinking if I could ...
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4answers
94 views

What is the reciprocal of $(-1/2)^k$?

What is the reciprocal of $(-1/2)^k$? The answer is meant to be $2^{-k}$ as if you flip something upside down the power becomes negative. However, I am not sure what happens to the negative in front ...
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1answer
36 views

How to compute $(a+1)^b\pmod{n}$ using $a^b\pmod{n}$?

As we know, we can compute $a^b \pmod{n}$ efficiently using Right-to-left binary method Modular exponentiation. Assume b is a prime number . Can we compute directly $(a+1)^b\pmod{n}$ using ...
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1answer
38 views

Audio frequency increment yielding wrong results

I am writing something in the ChucK programming language, which is designed specifically for audio time functions (in this case, hertz). I'm having a really difficult time with a mathematical ...
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0answers
100 views

The smallest non-zero integer $c$ such that $\sum\limits_{n=1}^m\pm(x+n)^6 = c$?

We have the neat equalities, I. Group 1 For $k=2,3,4,5,\dots$ $$\sum_{n=1}^{2^k}\epsilon_n(x+n)^k = 2^{\frac{k(k-1)}{2}}k! = 4,\;48,\;1536,\;\color{brown}{122880},\dots$$ for appropriate ...
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1answer
52 views

Can I simplify $ \ln(A/B)+C$ any more?

This should be a rather simple problem however I am having difficulty getting this simplified. If I need to simplify the expression $$ \ln(A/B)+C$$ My first step is $$ A/B + e^c$$ However MATLAB and ...
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4answers
58 views

How do you algebraically derive “x <= 0” from “-x = | x |”

A = "-x = | x |" B = "x <= 0" If A, then B. By plugging in numbers or testing ranges less than zero, greater than zero, and equal to zero, I can verify that A ...
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1answer
60 views

A problem with exponent laws…

Solve for $x$: $$x^{\frac 13}={32\over \sqrt{x}}$$ I'm not sure how start up this problem. I thought you had to multiply both sides by $\sqrt{x}$ so that it cancels out on the right side and moves ...
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0answers
38 views

How to reduce exponentiation expressions?

It is a simple question but I am afraid of its simplicity. Is that correct : $2^{30}+2^{30}+2^{30}+2^{30} = 2^{30}(1 + 1 + 1 + 1) = (2^{30})\cdot 4 = 2^{30}\cdot2^2 = 2^{32}$? I am doing complex ...
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2answers
28 views

Why is $-x = (x^2) ^ {\frac{1}{2}}$ not logically equivalent to $(-x) ^ 2 = ((x^2) ^ {\frac{1}{2}}) ^ 2$?

Why is $-x = (x^2) ^ {\frac{1}{2}}$ not logically equivalent to $(-x) ^ 2 = ((x^2) ^ {\frac{1}{2}}) ^ 2$ for all values of x? First equation: $-x = (x^2) ^ {\frac{1}{2}}$ Second equation: $(-x) ^ ...
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0answers
21 views

Help with repeated squaring

I'm having trouble figuring out how to use repeated squaring to figure out 289^377 mod 589. I've seen other websites break the exponent down into (1 + 4 + 16 ... ), but I'm not sure when to do that.
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0answers
59 views

Why can $x^0$ sometimes be simplified to 1 even when x can equal 0?

For example, the Taylor series for $e^x$ is $\sum_{n=0}^{\infty} \frac{x^n}{n!}$. It seems like it should be indeterminate or undefined at $x=0$, since the first term would contain $0^0$, but it's not ...
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0answers
38 views

Raising numbers with powers

So the question is as follows: Let $f(x) = \int_{0}^x \frac{x}{2\sqrt{t}}dt$. Suppose $f(f(f(...f(f(a))...)))$ (done $2013$ times) $= 2^{2013}$. Find the real-valued solution of $a$ Now, for my ...
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1answer
47 views

Pattern in Digits in Powers of 2

Along a similar line to this question, (pattern in decimal representation of powers of 5), I was playing around in a mathematics program called GAP. I was entering powers of two, when I noticed an ...
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1answer
47 views

How to simplify recurrence relation?

I'm having trouble seeing how $$5(2^{n-1} + 5\cdot 3^{n-1}) - 6(2^{n-2} + 5\cdot3^{n-2})$$ simplifies to: $$2^{n-2}\cdot (10 - 6) + 3^{n-2} \cdot (75 - 30)$$ How can I simplify the above ...
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1answer
27 views

Calculating the number of times a value must be halved for it to be less than or equal to another value

This is not a homework question; I'm working out an algorithm for an app I'm writing and I want to calculate the number of times I must halve a base value for it to be less than or equal to a minimum. ...
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1answer
60 views

Does (x^y)^a+b = x^(y*(a+b)) or x^((y*a)+b)?

I posses two Math books, both of which define a certain property of the algebraic manipulation of exponents in different ways. For example: Book one would claim that: 2^((3)2+3) = 2(3*5) = 2^15, ...
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1answer
61 views

An upper bound for $a_1 (\sum_{i=2}^n a_i^{n-1})$ in terms of $a_1^n + \sum_{i=2}^n a_i^n$

Assume that $n \in \mathbb{N}, n \geq 2$ and $a_i \in \mathbb{R}, a_i > 0, \forall i=1,...,n$ Show that $$ \frac{a_1 (\sum_{i=2}^n a_i^{n-1})}{ a_1^n + \sum_{i=2}^n a_i^n} \leq 1 -\frac{1}{n}$$ ...
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2answers
68 views

What is the correct value?

My confusion is: $(-9)^{2/3} = ((-9)^{2})^{1/3} = ((-9)^{(1/3)})^{2} = 4.32$ But my calculator shows math error, and google says: $(-9)^{2/3} = 2.16+3.74i$
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1answer
34 views

Simplifying exponents

I've been refreshing my maths over the last couple of weeks, and it's been a challenge since it has been a long time since I was actively using it (20+ years). Anyways, Khan Academy and old textbooks ...
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3answers
124 views

Is the solution of the equation $x^x=2$ rational?

Let $x$ be the solution of the equation $x^x=2$. Is $x$ irrational? How to prove this?
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0answers
75 views

Irrational numbers to irrational powers being rational?

So some of you may be familiar with the proof that some irrational numbers to irrational powers are rational, that is: if $A = \sqrt2^\sqrt{2}$ then it follows that $A^\sqrt{2} = 2$. So, I've found a ...
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1answer
79 views

Is there a $k$ such that $2^n$ has $6$ as one of its digits for all $n\ge k$?

It is true that every power of $2$ of the form $2^{6+10x}$, $x\in\mathbb{N}$, has $6$ as one of its digits. Something more is true, the last two digits are either $64$ or $36$. The OP suggests that ...
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1answer
45 views

Logarithmic question

In the following question I fail to understand why the A option is correct. I understand that D is wrong, and that B and C are correct, but why is A correct? If $3^x=4^{x-1}$, then $x $cannot be ...
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2answers
376 views

Prove an inequality on natural number

Show that if $ a,b\in N$ and $a < b$, then $$\frac{a^a}{(a+1)^{a+1}} > \frac{b^b}{(b+1)^{b+1}}.$$
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1answer
21 views

Find $E[W|X>Y]$ where $W = X+Y$ and $X,Y \sim \exp(2)$ independently

I need an idea on how to solve following conditional expectation $E[W|X>Y]$ where $W = X+Y$ and $X,Y \sim \exp(2)$ and $X$ and $Y$ are independent. Thanks.
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1answer
33 views

How to prove that $f(x) = x^ε - \log x$ is $\infty$ when $x\to\infty$?

I'm trying to prove that the function $x^ε$ is "bigger" than $\log x$ when $x\to\infty$, for every $ε>0$. Or to put it in a more formal way: For every $ε>0$, there exists a constant $N$ for ...
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0answers
57 views

How to solve an equation with absolute value and x as exponent.

The inequation is: I thought that the solution would be the same as if there was no x as exponent, but in Microsoft Math it says it has no solution and about the equation it said it has solutions. ...
2
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1answer
24 views

Properties of the exponent function attached to a nonzero prime ideal in a Dedekind domain

I want to prove properties of $v_\mathfrak{p}$, which I have been told is: "the exponent function attached to a nonzero prime ideal $\mathfrak{p}$ that maps a given nonzero fractional ideal to the ...
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0answers
34 views

A relation with limits

Let $t \in \mathbb{R}_+$, $\varepsilon > 0$ and $p_\varepsilon \in [0, 1]$. There is in a book the following relation: $$ \underset{\varepsilon \rightarrow 0}{lim} (1 - p_\varepsilon)^{\lceil t / ...
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1answer
7k views

why does e raised to the power of negative infinity equal 0?

Why is it that e raised to the power of negative infinity would equal 0 instead of negative infinity? I am working on problems with regards to limits of integration, specifically improper integrals ...
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3answers
45 views

Operations and Identities [duplicate]

We have the binary operation addition on numbers. It has an additive identity ( 0 ) and it is commutative. Multiplication is simply repeated addition. It is a binary operation on numbers. Its ...
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4answers
47 views

How to see that $2^{n-1} + 2^{n-1} - 1 = 2^n - 1$

How to see that $2^{n-1} + 2^{n-1} - 1 = 2^n - 1$? Is there a rule about adding two powers of the same base I'm not aware of? I know that you can "add the exponents" if you are multiplying numbers ...
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1answer
21 views

Proof that. for any two natural numbers $n$, $m$, $n^m$ is not an $n$-ary pernicious number.

$a_{10}$ is defined to be an $n$-ary pernicious number when the digit sum of $a_n$ is prime in base $10$. How can I prove that, for any two natural numbers $n$, $m$, $n^m$ is not an $n$-ary ...
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2answers
36 views

Implicit logarithmic differentiation to find the horizontal tangents of an exponential function

The graph of $y = 6{(3{x}^2)}^x$ has two horizontal tangent lines. Find equations for both of them. $$ \\ \begin{align} \\ y &= 6{(3{x}^2)}^x \\ y &= 6 \cdot {3}^x \cdot {x}^{2x} \\ \ln{y} ...
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1answer
41 views

Variable exponent solve for x

How can I solve this exponent problem using simple math only? We need to solve for $x$ $2^{2x}-(3.2)^{x+2} + 32 = 0$ The second term here is $3.2$ not $3\cdot2$ ie 3 decimal 2 not 3 into 2. My ...
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0answers
29 views

Simplified Exponent in $\bmod$ equation

I am trying to simplified the following expression: $$\left(y^m\right)^{x} \bmod p$$ In my case, I can only solve $(y^x) \bmod p$ first without prior knowledge of $m$. Eventually, my answer should ...
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2answers
41 views

Is it wrong to “imagine” a one when thinking about exponentiation? (e.g. $3^2 = 1 \times 3 \times 3$)

This might be a bit of a basic question, but I'm going through Khan Academy to refresh my math skills in order to pursue a self-study of higher mathematics, so I'm really focused on the "why" of the ...
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1answer
23 views

Multiplication Question with Powers

A quick question. I dont know how to do this. $$1-2^k\times 2+1\times (-1)$$
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11 views

Find distortion exponent from Fourier fitting

I'm facing this problem in my master thesis: we are measuring the signal from a sensor which is, physically, a $\sin^2$ (or $\cos^2$). Some non idealities distort the signal by introducing an exponent ...
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1answer
76 views

Besides $2^4$ and $4^2$, are there any other numbers that, with the base and exponent flipped, will equal the same value? [duplicate]

I've noticed that when you flip the base and the exponent in $2^4$ to get $4^2$, you get the same value, $16$. If there are any other numbers that can make this work, let me know. This is just ...
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0answers
37 views

exponent and modulus

Good day all, I am working on a equation and not getting it to work correctly. Hope to seek some advice. $r1 = (m+xr-m1)x^{-1}\pmod q$ $y^{r1} = (y^m)^{x^{-1}} \times y^{xr(x^{-1})} \div ...
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1answer
32 views

How can this exponential equation be computed

Is there a mathematical way to solve such equations, besides try and error of course? $e^{-x} = 1-x/5$
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1answer
69 views

Proving something about the sequence of powers of 3 mod 10. Oh boy

Given the sequence $t_0=3, t_1=3^3,...$ so that $t_{n+1}=3^{t_n}$, prove $t_{k+1} \equiv t_k \mod 10^n$ for all integers $n \leq k$ My work so far: I thought it was a pretty obvious case for ...
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3answers
21 views

Explanation for an equality involving complexed exponents

$$\frac{e^{i(N+1)x}-e^{-iNx}}{e^{ix}-1} = \frac{e^{i(N+1/2)x}-e^{-i(N+1/2)x}}{e^{ix/2}-e^{-ix/2}}$$ I'd be glad to get an explanation for both numerator and denominator. Thanks in advance!
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4answers
107 views

Find derivative of $x^{x^x}$

Trying to find the derivative of: $$ x^{x^x} $$ I have a solution but cannot understand the third transition:
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1answer
184 views

Prove that $1989\mid n^{n^{n^{n}}} - n^{n^{n}}$

Having difficulty in proving this: $1989\mid n^{n^{n^{n}}} - n^{n^{n}}$ for all $n \in \Bbb N$. Prime factorization of $1989$ is $3^2 \times 13 \times 17$. Please Help!
2
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3answers
140 views

Question regarding the square root of a squared number. [duplicate]

I've learnt that the square root of a number squared is equal to the absolute value of that number, but I haven't really understood why. I have looked through other questions on MSE but didn't really ...
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2answers
71 views

Prove that 100…500…1 (100 zeros in each group) is not a perfect cube?

How can i prove that 100...500...1 [100 zeros in each group ( ... is 100 zeros)]is not a perfect cube? I tried symmetric features of the number but could not figure out anything related.any ideas ...
3
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1answer
121 views

Arrangement of integers in a row such that the sum of every two adjacent numbers is a perfect square.

Inspired by this interesting question and in order to solve an old problem, I have the following question: Can we construct a strictly increasing sequence $(N_i)_{i\in \mathbb{N}}$, such that for ...