Questions about exponentiation

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0
votes
2answers
68 views

why is $(-64)^{2/3} =-16$ and not $16$?

It appears that taking the cube root of a negative number will yield a negative number, which when squared, will yield a positive number. But all the calculators and books I have seen show this ...
2
votes
2answers
594 views

Exponent rule and square roots?

For some $x$, $\sqrt{x^2} = |x|$ However, for $x= -1$. $\sqrt{(-1)^2} = (-1^2)^{1/2} = (-1)^{2/2} = (-1)^1 = -1$ Isn't this paradoxical?
0
votes
1answer
63 views

Is $\left((-1)^2\right)^\frac12 = (-1)^\left(2\cdot\frac12\right)$? [duplicate]

I'm feeling confused. If I square 1 and -1, the answers should be equal: $1^2 = (-1)^2$ Then I take both sides to the power of $\frac12$: $\left(1^2\right)^\frac12 = \left((-1)^2\right)^\frac12$ ...
1
vote
3answers
122 views

How do you calculate a large power modulo a small number? [duplicate]

How do I calculate $12345^{12345} \operatorname{mod} 17$? I cant do it on a calculator? How would I show this systematically?
0
votes
1answer
42 views

show that $(1+ \frac {x}{n})^n < e^x$ and $e^x < (1- \frac{x}{n})^{-n}$ if $x<n$

If $n$ is a positive integer and if $x>0$,show that $(1+ \frac {x}{n})^n < e^x \quad$ and that $\quad e^x < (1- \frac{x}{n})^{-n} \quad $ if $x<n$ I proved the first one by the ...
-1
votes
1answer
27 views

Rewrite a formula in terms of exponential to the power of logarithm

I would like to rewrite the following formula, f(x). how can I rewrite the f(x) $$ f(x) = ...
0
votes
2answers
73 views

Why is n^(1/m) no valid way to calculate a root

So I came across a situation where a calculator only had square root, but I needed the cubic root. So I used the old $n^\frac13$ trick, and sure enough, the cubic root of n. So this got me thinking. ...
18
votes
4answers
304 views

Find the smallest k such that $n^k > \sum_{i=0}^{n-1} i^k$

Let $n \in \mathbb{N}$. Is it possible to find the smallest $k \in \mathbb{N}$ such that $$n^k > \sum_{i=1}^{n-1} i^k \ ?$$ It's easy to prove that such $k$ exist because: $$n^k > 1^k + 2^k ...
2
votes
1answer
102 views

Modular exponentiation references

I have recently learned a trick in modular exponentiation that is new to me. By example (as in the linked question/answer above): $$2^{1386}=2^{2^{10}}\cdot 2^{2^8}\cdot 2^{2^6}\cdot 2^{2^5}\cdot ...
0
votes
2answers
27 views

Derivative: Which rule to use first?

$f(x)=x^7(5+8x)^3$ Would I go about finding the derivative of this problem by using the chain rule first, and then the product rule? Or would I do the opposite? Step by step instructions would be ...
3
votes
2answers
69 views

$\alpha \in \mathbb{R}$ and $2^\alpha$, $3^\alpha \in \mathbb{N}$, implies $\alpha \in \mathbb{N}$?

Let $\alpha \in \mathbb{R}$. Suppose $2^\alpha$, $3^\alpha \in \mathbb{N}$. Does it implies that $\alpha \in \mathbb{N}$?
1
vote
1answer
25 views

Comparing sum of fixed rate value to sum of escalating value

Find the number of years, $n$, until the sum of an escalating value/income exceeds the sum of a higher fixed level value/income. Income fixed at £8405.64 Income escalating @ 3% per annum from ...
5
votes
2answers
109 views

Prove that $\sqrt{8}=1+\dfrac34+\dfrac{3\cdot5}{4\cdot8}+\dfrac{3\cdot5\cdot7}{4\cdot8\cdot12}+\ldots$

Prove that $\sqrt{8}=1+\dfrac34+\dfrac{3\cdot5}{4\cdot8}+\dfrac{3\cdot5\cdot7}{4\cdot8\cdot12}+\ldots$ My work: $\sqrt8=\bigg(1-\dfrac12\bigg)^{-\frac32}$ Now, I suppose there is some "binomial ...
4
votes
3answers
49 views

Graph of the last digit of $x^n$ - why is it symmetric when $n$ is even, and not when $n$ is odd?

I have discovered this fact: "The graph of the last digit of $x^n$ (where $x$ is positive) is asymmetrical if $n$ is odd, and symmetrical if $n$ is even." What is the logic behind this? For ...
0
votes
2answers
27 views

Fractional Power Interpretation

I have a following query in my mind. It has been in my mind since i was a kid. I know that 2^3 means that multiply 2 three times,3^-2 means multiply (1/3) two times.What does 2^(0.22) means. multiply ...
2
votes
2answers
46 views

$2^n$ modulo n where n is odd always yields either an even or $1$

I'm attempting to do a pidgeonhole proof to prove that for some odd integer n, there is always a $2^k$ such that $2^k \mod(n) = 1$. I know that $2^n \mod(n)$ will always yield either an even number or ...
0
votes
0answers
16 views

compounding interest question

A bank advertises that it compounds interest continuously and that it will double your money in 7 years. What is the annual interest rate? P(t) = P*e^kt P(t)/P =2 e^7k = 2 take ln of Both Sides ...
1
vote
2answers
25 views

Continuous compounding question

A population of rabbits starts out with $100$ rabbits. The growth rate is $11.7$% per day. Determine the exponential equation. Is it $$\mathbb {P(t)} = 100e^{11.7t}$$ Can you guys give me the ...
0
votes
5answers
80 views

How to solve $5^{n+2} - 5^{n-3} = -2500$ [closed]

How to solve $5^{n+2}- 5^{n-3} = -2500$
0
votes
2answers
110 views

Show that $\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{a} =1$

Hi guys this was a practice problem I was given, can anyone help me out on it? This is the problem: Show that $\displaystyle \lim_{n \rightarrow \infty} \sqrt[n]{a} =1$ and the following is what I ...
-1
votes
4answers
146 views

Does $\infty^0=1$?

I was wondering if $\infty^0=1$. Some people have told me that there is no answer; it is undefined. Others have told me that the answer is $1$, using the rule $a^0=1, \ a\neq 0$. If it is truly ...
0
votes
1answer
43 views

Exponential of $\bar{z} $

I am currently reading the book Complex Variables by Stephen Fisher, there is one paragraph that was written like this: Establishing the following relation, and they write ...
1
vote
0answers
72 views

Rising/Falling Powers, Summation

1) Show that $$(-n)^{\bar p} = (-1)^n n^\underline{p}$$ (original screenshot) 2) Evaluate the sum $$\sum_{a\le n\lt b}n^{\bar p}$$ (original screenshot) Thoughts regarding question 1: I've ...
3
votes
2answers
95 views

Limit of $\frac {n^n}{n!}$ [duplicate]

I have to prove that $$\lim_{n\to \infty} \frac {n^n} {n!}=\infty$$ I've tried to look for a lower bound that also converges to $\infty$ (I don't know if I'm explainig myself correctly), but I ...
2
votes
2answers
166 views

Solve exponential-polynomial equation

Solve the equation in $\mathbb{R}$ $$10^{-3}x^{\log_{10}x} + x(\log_{10}^2x - 2\log_{10} x) = x^2 + 3x$$ To be fair I wasn't able to make any progress. I tried using substitution for the ...
0
votes
2answers
58 views

Why is X raised to the power of 0 = 1?

So as the topic states, why is 5^0 = 1 and not 5 or 0? Is it only because the other exponential laws wouldn't work if it was not the case?
0
votes
0answers
22 views

How large does $m$ have to be to get unique values with high probability? [duplicate]

We can suppose we are given two naturals, $r$ and $n$. We can then pick $n$ unique naturals: $\{x_0, x_1, \dots, x_n\}$. The following function is important: $$\prod_{k=1}^n{(x_k)^{y_k}} (\mod m)$$ ...
-1
votes
1answer
40 views

How to solve the integral $\int x\cdot 9\cdot x^{2x^2} dx$

How to solve the integral $$\int x\cdot 9\cdot x^{2x^2} dx$$ I tried $u=2x^2$ and $du= 4x\;dx\Longrightarrow$ $$\int x\cdot9\cdot x^{2x^2}\;dx=\frac94\int x^u du$$But it was unable to pass.
3
votes
4answers
134 views

Computing a large exp(x) in a numerically robust way.

I'm trying to compute $\lfloor e^x \rfloor$, where x is a 64-bit integer. The problem is that the result of the computation may be close to 2^64. In this range, 64-bit floating point numbers will be ...
0
votes
2answers
69 views

Converting powers of 3 into powers of 2? [closed]

I'm stuck on a problem. I have a term $3^m$ where $m$ is an integer $> 0$. and I want to represent it as $2^m + a$ however I don't want to keep the $a$. I am looking for a formula to represent $a$ ...
2
votes
2answers
77 views

How hard is finding values such that

We can work with powers of some naturals $(x_k)^{m_k}$. Here we have $n$ naturals, and $m_k$ is an integer in the range $-r$ to $r$. My question is, how small can $p$ be so that ...
0
votes
1answer
30 views

About definition of $a^b$ with $a \in \Bbb{R} \wedge b \in \Bbb{N}$

-- let $a,c \in \mathbb{R}$, and $b \in \Bbb{N}$, with $\Bbb{R}$ is a complete ordered field, $c \triangleq a^b$ if $c=\begin{cases} 1, & \mbox{if } a\neq 0 \wedge b=0\\ 0, & \mbox{if } a=0 ...
2
votes
3answers
88 views

$n$-abelian Groups

Show that $(xy)^n=x^ny^n$ if $xy=yx$. I assume I will need 3 different cases: $n < 0$, $n=0$, and $n > 0$. For the $n > 0$ case, can I use induction? For the base case I'll show that ...
2
votes
0answers
41 views

Find a holomorphic function that satisfies this equation

Let $f:\mathbb{C}\times\mathbb{C}\rightarrow\mathbb{C}$ denote a function that satisfies the following equation: $$f(w,z+1)=w^{f(w,z)}$$ Is there a unique holomorphic function $f$ that satisfies this ...
0
votes
3answers
49 views

How do I compute the individual terms of a polynomial to the power of -1?

If my polynomial $p$ is: $x+1$, obviously $p^{-1} = \frac{1}{x+1}$. Is it possible for me to split $\frac{1}{x+1}$ into a sum of two terms? In other words, is there an algorithm to write $p^{-1}$ as ...
0
votes
1answer
95 views

How do I solve $|\sinh(x+iy)|^2 = (\sin(y))^2+(\sinh(x))^2$

How do I solve this ? $|\sinh (x+iy)|^2 = ( \sin (y))^2+ ( \sinh (x))^2$ I'm not sure how to solve the left hand side.
5
votes
6answers
237 views

Why is exponentiation defined as $x^y=e^{\ln(x)\cdot y}$?

There are many curves that extend integer exponentiation to larger domains, so why was this one chosen?
2
votes
2answers
105 views

Is it possible to solve $i^2+i+1\equiv 0\pmod{2^p-1}$ in general?

While looking at the Mersenne numbers (for prime $p$, the number $2^p-1$), I noticed that only certain of them had any solution to the modular equation $i^2+i+1\equiv 0\pmod{2^p-1}$, e.g., ...
1
vote
2answers
45 views

modular exponentiation where exponent is 1 (mod m)

Suppose I know that $ax + by \equiv 1 \pmod{m}$, why would then, for any $0<s<m$ it would hold that $s^{ax} s^{by} \equiv s^{ax+by} \equiv s \pmod{m}$? I do not understand the last step here. ...
3
votes
1answer
59 views

A question on Exponential Equation

I came across the following question a few week ago (Exponential equation+derivative): Solve $3^x+28^x=8^x+27^x$. The answer for the above question is 0 and 2. I generalized the question, as ...
0
votes
3answers
62 views

Logarithm properties doubt

The problem is $\log (5.64)^4$. According to the properties and laws of exponents, $\log (m^r) = r \log (m)$. But since the exponent is outside of the parenthesis in this problem, does it solves by ...
4
votes
1answer
57 views

Is $n^{\log c} = c^{\log n}$ true?

Is $n^{\log c}$ the same as $c^{\log n}$? If so, please explain.
1
vote
2answers
146 views

How to find out the value of numbers having fractional powers

How to find out the value of numbers having fractional powers manually without using logarithms and calculators?? For example : $2^{1.6}, 3^{2.1}, 5^{3.22}$ etc, I know we can find out the value using ...
0
votes
1answer
27 views

Is there a seperate something in front logarithm that is raising a base to a power?

I am trying to solve a problem with the following form $$e^{\displaystyle A\log(x)}$$ $e^{\log(x)}$ is simply $x$, but how do I go about separating the $A$?
2
votes
2answers
153 views

x raised to itself infinite number of times [duplicate]

$$\Large x^{x^{x^{x^{x^{.^{\,.^{\,.}}}}}}} = 2$$ What is $x$?
1
vote
1answer
24 views

Proof for exponentiation in modular arithemetic

I have found out, that the following is true for modular arithmetic when $t$ is a natural number. $$a^t \bmod\ n \equiv (a\bmod\ n)^t\bmod\ n$$ But I have been unable to find a proof for this, does ...
10
votes
1answer
181 views

Is $\exp:\overline{\mathbb{M}}_n\to\mathbb{M}_n$ injective?

More specific to my problem, this is a variation on Is $\exp:\mathbb{M_n}\to\mathbb{M_n}$ injective? which was promptly answered with a counterexample. Let $\mathbb{M}_n$ be the space of $n\times n$ ...
3
votes
2answers
71 views

Is $\exp:\mathbb{M_n}\to\mathbb{M_n}$ injective?

This is related to a personal exploration of isometries of directed graphs, motivated by my son's Lego Duplo train tracks and identifying "interesting" layouts. If $M$ is the adjacency matrix for a ...
0
votes
0answers
49 views

Are the solutions to this integral known?

Mathematica knows that the real part of $y$ in this integral: $$\int_0^{\infty } \frac{1}{x^{1/y}+2} \, dx$$ is: $$0<\Re(y)<1$$ Therefore I am wondering if the solutions to this integral known? ...
0
votes
0answers
29 views

Generalization for a power of an infinite summation

I am trying to obtain a simplification for the following expression: $$ \left[\sum_{m=0}^{\infty} \frac{\mu^m \kappa^m}{m! \Gamma\left(\mu+m\right)} \alpha^{-(\mu+m)} \Gamma\left(\mu+m, ...