Questions about exponentiation

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0answers
31 views

How can I simplify the following expression with exponents.

$$\frac{(t+1)^{\frac{1}{3}}-\frac{1}{3}t(t^2+1)^{-\frac{2}{3}}}{(t^2+1)^{-\frac{2}{3}}}$$ I found this problem from a book and its answer is $\frac{2t+3}{3(t+1)^{\frac{4}{3}}}$(as in the book's ...
0
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1answer
24 views

Name of numbers in “to the power of” and factorial calculations

In $4*5=20$ , $4$ and $5$ are multiplicands and $20$ is the product. What are the names / labels of the numbers in the following expressions? $2^3=8$ $4!=24$
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2answers
36 views

How to solve an exponential function with multiple addends

Our math teacher gave us the following exponential equation to solve: $3^x+10=2*7^x$ ...and I was stumped. Eventually, the solution given was to graph both sides and find their intersection using a ...
7
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1answer
238 views

Exponential of a function times derivative

Exponential of a derivative $e^{a\partial}$ is simply a shift operator, i.e. \begin{equation} e^{a\partial}f(x)=f(a+x) \end{equation} This can be easily verified from a Taylor series \begin{equation} ...
21
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8answers
2k views

How does the exponent of a function effect the result?

The $x^{2/2}$ can be represented by these ways: $$\begin{align} x^{2\over2}=\sqrt{x^2} = |x|\\ \end{align} $$ And $$\begin{align} x^{2\over2}=x^{1} = x\\ \end{align} $$ Which one is correct? And what ...
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1answer
50 views

Why 1.66x10e-27 is 1.66e-26

I am confused with this new way of writing exponential. For example I saw a question where person has changed 1.66X10e-27 to 1.66e-26. I am confused how -27 is reduced to -26 thanks
3
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1answer
79 views

Is there any way to analyze an absurdly large exponent?

On a recent Giant Bombcast, someone wrote in and asked an absurd question (as is usual for this podcast). In short, the question was: Given a 1080p TV, how long would it take to view every ...
2
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1answer
39 views

Permuting digits in a power of $2$

Does there exist a natural number $N$ that is a power of $2$ whose digits (in the decimal representation) can be permuted to a different power of $2$? Thoughts: If such a number $N$ exists, then ...
3
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2answers
71 views

Exponents with the same power

I've wanted to practice solving simple operations on exponents, so I've made a couple of equations to which I know the answers. $$5^x -4^x = 9$$ I feel really stupid, because I can't solve this one ...
0
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0answers
29 views

Trigonometric Power Formulas (or something more modest)

How does one begin to show (natural $n$): $$\cos^{2n}(x) =\frac{1}{2^{2n}} \binom{2n}{n}+ \frac{1}{2^{2n-1}} \sum_{k=0}^{n-1} \binom{2n}{k} \cos[2(n-k)x]$$ $$\cos^{2n+1}(x) =\frac{1}{4^{n}} ...
13
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5answers
431 views

Value of $(-1)^x$ for $x$ irrational

I was working on an analysis problem when this question arose in one my proofs. I think it may be either $-1$ or $1$, but it seems like there can only be an arbitrary way to assign this. So is there ...
0
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3answers
65 views

$(5^{2x}-1)(5^x)=1/5^x$ solve

I have the problem $(5^{2x}-1)(5^x) = 1/5^x$. I have already simplified it to $5^{3x}-1=1/5^x$ My question is when I do $\log$ base $5$ to the left side of the equation to get $3x-1$ by itself so ...
1
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1answer
50 views

moving trig function from the exponent?

Is there a form of $e^{\sin(x)}$ that does not have trig functions in the exponent? I've seen the "Euler Identity" form and looked into series expansions. (thanks for the answers -- sorry I didn't ...
0
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0answers
67 views

Complex exponentiation

So I've got this question that is a bit difficult to ask, since it uses a term in my language that I can't properly translate into English. For $z\in\mathbb{C}^*$ and $a\in\mathbb{C}$ it would be ...
4
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2answers
106 views

$2 \uparrow^n 2 = 4$ and the magnificence of $2$

I was reading up on tetration when I realized: $$2 \uparrow\uparrow 2 = 2\uparrow 2 = 2 \times 2 = 2+2 =4$$ Infact, when generally speaking: $$ 2 \uparrow^n 2 =4$$ Now, I realize that this is because ...
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3answers
70 views

Intricate exponential equation

This is the question: $$ \frac{(2^{3n+4})(8^{2n})(4^{n+1})}{(2^{n+5})(4^{8+n})} = 2 $$ I've tried several times but I can't get the answer by working out.I know $n =2$, can someone please give me some ...
0
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0answers
15 views

How would I design a formula that increases the length of pauses exponentially based on current speed?

I'm writing a program that presents users words in a flash-card fashion, at a speed they define (say, 500 cards/min). When a "section" of cards is done, I want there to be a pause before the next one ...
-2
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1answer
106 views

why e raised to power 1 is 2.71.. [closed]

Please can somebody explain and interpret why e raised to power 1 is 2.71 and from where did it come around in mathematics? since it has become vital in mathematics and science.
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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.
13
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1answer
192 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 $z_1, z_2\in \mathbb{C}, {z_1}^{z_2}=e^{z_2\ln(z_1)}$ and $\ln(z_1$) can just be found using: ...
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1answer
74 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 ...
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1answer
35 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?
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10answers
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 ...
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2answers
89 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? ...
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3answers
56 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
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1answer
54 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
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2answers
129 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
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7answers
129 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
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3answers
144 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
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1answer
114 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
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2answers
98 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
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2answers
42 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
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2answers
69 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
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2answers
660 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?
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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 $\frac12$: $\left(1^2\right)^\frac12 = \left((-1)^2\right)^\frac12$ ...
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3answers
123 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?
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1answer
43 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 ...
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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
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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
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4answers
306 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 ...
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1answer
103 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
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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
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2answers
71 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
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1answer
26 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
110 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
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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
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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
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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
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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 ...
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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 ...