Questions about exponentiation

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0
votes
1answer
19 views

Simplify this indices?

Simplify this: $6a^3 * {a^{-5}\over2}$ I got $6a^3 * {1\over2a^5}$ What should I do next? please explain with steps.
11
votes
1answer
102 views

How to raise a number to a quaternion power

Now, I know that it's (relatively) easy to calculate, say, $r^{a+bi}$ (using the fact that, for $z_1, z_2\in \mathbb{C}, {z_1}^{z_2}=e^{z_2ln(z_1)}$ and $ln(z_1$) can just be found using: ...
0
votes
1answer
54 views

another question on surds and how to use math symbols in this site

$$\sqrt{ 3x }= x + \sqrt {3}$$ this is what i tried $$\sqrt{ x }= (x + \sqrt {3})^2\\ = x^2 + 3 $$ Give x in the form $$A \sqrt {B} + C $$ Can you show me how this is done step by step. The ...
0
votes
1answer
30 views

Need help with self study on surds

give x in the form A $ \sqrt B$ + C $(\sqrt 3x) = 3$ can someone show me how to solve this please?
41
votes
9answers
2k views

What is exponentiation?

Is there an intuitive definition of exponentiation? In elementary school, we learned that $$ a^b = a \cdot a \cdot a \cdot a \cdots (b\ \textrm{ times}) $$ where $b$ is an integer. Then later on ...
1
vote
2answers
42 views

Matrix Exponential of Identity Matrix

I was just wondering what would the sum be of $e^{I_n}$ where $I_n$ is the identity matrix. I know the maclaurin series for $e^x$ is $1+\frac x{1!}+\frac {x^2}{2!}+...$. I know that $e^0$ is 1 right? ...
1
vote
3answers
54 views

If $(x^y)^z = x^{y\cdot z}$, why does $(-5)^{2^{0.5}}$not equal $(-5)^1$?

As shown by Wolfram Alpha, $(x^2)^{0.5}$ is equal to |x|, but if you tried to simplify it to $x^{2\times {0.5}}$, it would just be $x^1$, or $x$. Is there some unwritten rule about that distribution ...
2
votes
1answer
52 views

$|x|^{|x|}$ is continuous in $\mathbb{R}$

I'm trying to show this now my self, but still no go. There isn't really a concrete attempt that I can show.. Help?
4
votes
2answers
125 views

Last digits don't change when exponentiating

While playing around with Wolfram Alpha, I noticed that the last four digits of $7^{7^{7^{7^7}}}, 7^{7^{7^{7^{7^7}}}},$ and $7^{7^{7^{7^{7^{7^7}}}}}$were all $2343$. In fact, the number of sevens did ...
2
votes
7answers
116 views

for $n$ an integer, why is $n^0=1$ ??

This is so going to cost me.... I was wondering why for any integer $n$: $n^0 =1$. Perhaps It's because $n$ is a round number and if $m$ is a non negative integer, also round then: $$n^m = 1 \cdot ...
2
votes
3answers
104 views

Lie group, Lie algebra and surjectivity

Let $G$ be a connected Lie group. If the Lie algebra $\mathfrak{g}$ is commutative, is the exponential mapping surjective? If not, do we at least have that $G$ is abelian? Any counter-examples as I ...
0
votes
1answer
57 views

Approximating Logs and Antilogs by hand

I have read through questions like Calculate logarithms by hand and and a section of the Feynman Lecture series which talks about calculation of logarithms. I have recognized neither of them useful ...
4
votes
2answers
78 views

How to resolve a power of a negative number?

$\left(-64\right)^{\left(\frac{3}{2}\right)}$ (Disclaimer - I work in a HS math center, helping students. This is from an Algebra/Trig text used by both sophomores and juniors depending on the class. ...
2
votes
2answers
37 views

Reducing exponents with a common base when terms are added

I have a series of terms as follows: $$e^{6x\pi.0} + e^{6x\pi.2} + e^{6x\pi.4} + e^{6x\pi.6}$$ Obviously the first term is just 1 but is there a way to specify the terms in one single term or ...
0
votes
2answers
63 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
332 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
62 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
94 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
37 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
20 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
67 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
284 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 ...
0
votes
0answers
22 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
25 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
67 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
19 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
106 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 ...
0
votes
0answers
50 views

What is the integration of the following definite integral?

$$I = \int_{\pi /2}^{(M - 1)\pi /M} \left(\frac{A}{B + \frac{C}{\sin^2\theta}} \right)^D \exp\left(\frac{A}{B + \frac{C}{\sin^2\theta}} \right)d\theta$$ I tried to do by putting $$t=\left(\frac{A}{B ...
4
votes
3answers
43 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
26 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
45 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
14 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
24 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
76 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
99 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
143 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
37 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
61 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
84 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
1answer
79 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
45 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
39 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
117 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
65 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
74 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
27 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
78 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
40 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 ...