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

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13 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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1answer
47 views

Complex exponential: how does $e^{-x} = e^{-1}$?

In the complex space how does: $$|e^{-z}| = e^{-x} = e^{-1}$$ I understand why it is only the real part, it is the step from $e^{-x} = e^{-1}$, where I am stuck. edit: this has to do w.r.t the ...
4
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0answers
55 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
34 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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2answers
35 views

Exponent from 3 points on an x-y graph [closed]

Given $3$ points plotted on an x-y graph, is it possible to determine the exponent? And if so, what is the equation? The $3$ points are as follows: $(-0.22, -0.45),(-0.13, 0.68),(0.31, 0.86)$
2
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1answer
31 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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0answers
36 views

What expression would one use to express the following? [closed]

130 - 120 = -1.39 120 - 110 = -1.51 110 - 100 = -1.65 100 - 90 = -1.82 90 - 80 = -2.04 80 - 70 = -2.31 70 - 60 = ? What expression expresses the above? I think it's exponential.
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1answer
30 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 ...
1
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1answer
23 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. ...
0
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1answer
38 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
60 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 $$ \frac{a_1 (\sum_{i=2}^n a_i^{n-1})}{ a_1^n + \sum_{i=2}^n a_i^n} \leq 1 -\frac{1}{n}$$ ...
2
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2answers
56 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: $(-9)^{2/3} = 2.16+3.74i$
3
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1answer
29 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 ...
5
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1answer
67 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 ...
0
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1answer
40 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 ...
6
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2answers
360 views

Prove an inequality on natural number

Show that if $ a,b\in N$ and $a < b$, then $$\frac{a^a}{(a+1)^{a+1}} > \frac{b^b}{(b+1)^{b+1}}.$$
0
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1answer
19 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.
1
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1answer
32 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 ...
0
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0answers
23 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
19 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 ...
2
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0answers
28 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
93 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
36 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
45 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 ...
1
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1answer
17 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 ...
1
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2answers
25 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
34 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
28 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
38 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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0answers
9 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
64 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
24 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
31 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
54 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
89 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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3answers
73 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 ...
1
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2answers
53 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
81 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 ...
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0answers
17 views

A more general class for power-law distribution

Is a power-law distribution, also a: Fat-tailed distribution? Heavy-tailed distribution? Long-tailed distribution? Also, which of the three distributions above is a subclass of the other?
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1answer
458 views

Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power

I've known that one can arrange all the numbers from $1$ to $\color{red}{15}$ in a row such that the sum of every two adjacent numbers is a perfect square. $$8,1,15,10,6,3,13,12,4,5,11,14,2,7,9$$ ...
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2answers
50 views

Differentiate $f(x) = exp_{a}(x) $ from first principles

Differentiate $f(x) = exp_{a}(x) $ from first principles, for $ a > 0 $ (Recall that $ exp_{a}(x) = exp(x.ln(a)) $ Here is where I am so far: $ f'(x) = \lim\limits_{h \rightarrow 0} ...
0
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1answer
46 views

Solving for $z^2 = x^2 -xy + y^2$

Recently, I came across the following solution to finding integer solutions for $z^2 = x^2 - xy + y^2$: $x = k(-n^2 -2mn)$ $y = k(m^2 - n^2)$ $z = k(mn + m^2 + n^2)$ I've been scratching my head ...
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2answers
45 views

Modular Arithmetic with Multiple Exponents

I understand how to do modular arithmetic on numbers with large exponents (like $8^{202}$). However, I am having trouble understanding how to calculate something like: $ 3^{3^{3^{3^3}}}$ mod 5 ...
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5answers
45 views

How to use the chain rule to differentiate

I have two examples of problems that I don't know how to differentiate. y = $e^{x^2/3x+2}$ and y = $-10x^{3x^2-4}$ I know to take the ln on both sides. I just don't understand whereto go ...
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3answers
30 views

A question in powers

I am a student in the second secondary so I had a question in math about powers like 4 power what gives you half in mathematical way I mean if there is a rule or a theory which can help me if I fall ...
0
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1answer
32 views

Modular Congruence, Power

$(3x^2+2x+1)(4x^3+x^2+5x+1)=5x^5+4x^4+1\equiv 1 \pmod {x^4}$ Expanding the first part, I get $12x^5+11x^4+21x^3+14x^2+7x+1$. However, I do not understand how to get from the above statement to ...
0
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0answers
53 views

Assuming $m \ge 3$, under what conditions is $z^{m-1} = \frac{x^m + y^m}{x + y}$?

One case where this is true is $z=1$: $$(1)^{m-1} = \frac{1^m + 1^m}{1+1}$$ Another case is where $x=y$: $$(x)^{m-1} = \frac{x^m + x^m}{x+x}$$ Assuming that $x,y,z,m$ are positive integers with $z ...
1
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0answers
65 views

Reasoning about $z^n = x^m + y^m$

Let $z,n,x,y,m$ be positive integers with $z \ge 5$ and $m \ge 3$ and $m$ odd. Does it follow that: $z$ cannot be prime if $p \ge 5$ and $p | z$, then either $p > m$ or $p|m$ Here is my ...