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

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4answers
28 views

Find the derivative of y with respect to the given independent variable

Find the derivative of y with respect to the given independent variable: $y = 3^{-x} \stackrel{D}{\longrightarrow} y' = 3^{-x} \cdot (-1) \cdot \ln 3 $ This is my teacher's solution. I don't ...
2
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0answers
201 views

Find the value of a^b?

Given a series as f(n)=1^1 * 2^2 * 3^3 * ......n^n since n can be very large Find the value of f(n)/f(r)*f(n-r) and output it modulo m where m is any prime. Now My approach is f(n)=1^1%m * ...
2
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1answer
37 views

How to deal with negative exponents in modular arithmetic?

So I think I understand how to calculate something like $(208\cdot 2^{-1})\mod 421$ using extended euclidean algorithm. But how would you calculate something like $(208\cdot2^{-21})\mod 421$? ...
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1answer
23 views

Getting rid of a fractional power over f(x)

I start with the following relation. $\frac{dy}{\sqrt{y}} = -h dt$ I then integrate it and get this function. $y^{\frac{3}{2}} = -\frac{3}{2}ht + C$ My algebra is rusty, so I'm stuck at this ...
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0answers
45 views

A variant of factorial

Given the definition of a function f as f(n)=1^1 * 2^2 * 3^3 * ... * (n-1)^(n-1) * n^n. Another function g is defined as g(n,r)=f(n)/(f(r)*f(n-r)) Given an n,r,m we are to output g(n,r)%m where m is ...
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3answers
44 views

Real analysis of powers

Show that if $a,b$ are rational numbers and $x$ is a positive real number then $x^a$$x^b$ $=$ $x^{(a+b)}$ I honestly have no idea how to even do this. Anyone have any hints or a good explanation? ...
3
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1answer
22 views

$y = ln(p+qe^x)/x$, solve $x$

$y = \ln(p+qe^x)/x$ $p$ and $q$ are constants. Express $x$ in terms of $y$. I believe I have to use Lambert W function, but I'm stumped. Thinking help is needed. Thank you very much!
2
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2answers
40 views

Why does zero raised to the power of negative one equal infinity?

I had the question of $0^{-1}$ on a math test and I naturally assumed that this evaluates to zero, but from what I have seen from various sources it is equal to infinity which I do not quite ...
2
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0answers
38 views

When is iterated exponentiation used and how is it defined?

I was thinking of ways to define an iterated exponentiation operation. The nice thing about addition and multiplication is that they're associative and commutative, which makes defining the sum and ...
2
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5answers
48 views

How to solve large modulus manually? [closed]

given $52^{35}\mod 85$ I know I can change that to $52\cdot 52^{2^{16}}\mod 85$ but I'm unsure where to go from there.
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0answers
19 views

Limit of power function

I want to be sure if that is correct, Could you please point me to some good references for that kind of function : When $a \to +\infty$ $ x^a =\begin{cases} 1 & x = 1\\ \\ +\infty & x>1 ...
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0answers
27 views

How are the power of 2 and square roots consistent?

This is something that keeps bothering me. We all know that $x^{2} = |x|^2$ And we know that, e.g., $\sqrt{9} = 3$ But we can also use $\sqrt{x} = x^{\frac{1}{2}}$ and $(x^{a})^{b} = x^{ab}$ ...
0
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1answer
31 views

Periodocity of $a^{pn+q}$ mod $m$

Is $a^{pn+q}$ mod $m$ periodic? $a$, $p$ and $q$ are constants. $n$ is varied here. If it is periodic then how can I find the periodicity efficiently? Thanks in advance.
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0answers
39 views

Commuting exponentials of non-commuting matrices

For two non-commuting matrices $A,B \in M(2,\mathbb{K})$, $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$, can be shown that: $$ e^C=e^{A+B}=e^Ae^B=e^Be^A \iff \begin{cases} ...
0
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1answer
18 views

Converting to Power of Ten Representaion

For very large calculations Wolfram Alpha offers a variety of different representations of the number. One of these is the number written in the form $10^{10^n}$, where $n$ is usually some long ...
1
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1answer
69 views

Quaternion exponential

Given an imaginary quaternion $ \mathbf{v}=\alpha \mathbf{i}+ \beta \mathbf{j}+\gamma \mathbf{k} $ its exponential is: $ e^\mathbf{v}=\cos \theta +\mathbf{v}\dfrac {\sin \theta}{\theta} $ where ...
2
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1answer
49 views

What's bigger, the sum of powers or the power of the sum?

Do we know if $(\sum\limits_{i=1}^n a_i)^k \geq \sum\limits_{i=1}^n a_i^k$ for any $k\geq1$?
2
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3answers
81 views

Solving $\ln(x) = e^{-x}$

I'm trying to solve $\ln(x) = e^{-x}$ but I can't really get how to do it :( (Removing a statement that was incorrect, as explained by the comments below) Additionally, while I started to solve it I ...
0
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1answer
53 views

rational exponent of negative base

I have the definite integral $$\int_{1}^{\,9} {\frac{6}{\sqrt[3]{x-9}}}\, \mathrm dx$$ When I try to evaluate it I get the indefinite integral equals $9(x-9)^{2/3}$ and evaluating at the limits gives ...
1
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2answers
45 views

Rules of i ($\sqrt -1$) to a power

$i^{2014}$ power =? A. $i^{13}$ B. $ i ^{203}$ C. $i^{726}$ D. $i^{1993}$ E. $i^{2100}$ I don't understand the concept that powers of i repeat in fours and that "two powers of i are equal if ...
0
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0answers
47 views

A basic doubt to compute exponential of a matrix

Given a matrix I want to evaluate $e^{A}$. The method suggested uses the taylor expansion. But, it is also written that the method works well if the largest and smallest eigen values are not well ...
1
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0answers
60 views

Are exponents with a base very close to $1$ (such as $1.0001$) useful in Mathematics?

I was curious if exponents with a base very close to $1$ are ever used in Mathematics and for what applications. For example, when I was in college, my Calculus professor told me that logarithms are ...
2
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2answers
24 views

Negative imaginary exponents

I was reading this question earlier: Understanding imaginary exponents In the answer, the answerer says $A^i=x+iy$ Furthermore, we can write $A^{−i}=x−iy$ for the same $x$ and $y$. Can ...
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3answers
113 views

Why does 2^-2 equal 1/2^2?

My intuitive way of thinking about it is that it is $2/2/2$ or $2/2^2$, So why then is it $1/2^2$? what is the flaw in my thinking?
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3answers
41 views

How do I figure out the value of a number raised to a fractional power? [duplicate]

For example, if I had something like $9^{1\over 2}$, how should I determine how many times I should use the base (the number being raised to the power of the exponent) as a factor? I just need ...
0
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0answers
30 views

Elementary Analysis, 3rd root question

Prove that $\forall a \in \mathbb{R}$ there is a unique solution to $x^3 = a$ Prove that $\forall x,a \in \mathbb{R}$ $$(x^{1/3}-a^{1/3})(x^{1/3})^2 + a^{1/3} x^{1/3} + ((a^{1/3})^2)=x-a$$ Prove ...
0
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1answer
49 views

Is there a such a thing as a “log root” or perhaps a “power root”?

We all know what a square root is. It's the number that for any $x$, you can calculate $n * n = x$. Is there an equivalent function to determine for any $x$ that you can calculate $n ^ n = x$? How ...
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0answers
33 views

Golden Ratio Sandbox

This might be a little long so please bear with me. The Golden Ratio $\phi$ is defined as the single positive root of the polynomial $p(t) = t^2 - t - 1$. One can think of it as a line divided into ...
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2answers
34 views

derivative of $\frac{2}{3}x^{3-e}$

Find the derivative:$\;\;\;\;\;\;\dfrac{2}{3}x^{3-e}$ I am not sure how to solve this problem. My try: $\ln y=\dfrac{2}{3}(3-e)\ln x$ $\dfrac{1}{y}\times y\;'=\dfrac{2}{3}(3-e)\dfrac{1}{x}$ ...
2
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0answers
28 views

What does it mean to have an irrational/imaginary exponent and is there a way to calculate the latter?

In exponentiation, we are told that raising something to an integral power (n, say) means multiplying it with itself a total of n times, if n is non-negative. And we also learn fairly early on that ...
1
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3answers
20 views

Evaluating the value of exponential expression

What is the value of: $\frac{2a}{a^{x-y}-1}+\frac{2a}{a^{y-x}-1}$ I tried this: $(\frac{a^{x-y}-1}{2a})^{-1}+(\frac{a^{y-x}-1}{2a})^{-1}$ $\frac{({a^{x-y}-1})^{-1}+({a^{y-x}-1})^{-1}}{2a^{-1}}$ ...
0
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2answers
50 views

Difficulty understanding addition of exponents

If you rewrite $(3^{12})(3^{-12})$ in the form $3^n$ what does it equal? What is the intuition behind it? Do exponents cancel each other out so it is just $3$? or do the negatives cancel out $((-x) ...
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2answers
38 views

derivative of $e^{\ln x^2}-3x^7$

$$e^{\ln x^2}-3x^7$$ The first term: $=e^v$ $v=\ln x^2=u^2$ $v\;'=2uu\;'=(2\ln x)\dfrac{1}{x}=\dfrac{2\ln x}{x}$ $\dfrac{e^{\ln x^2}2\ln x}{x} +21x^{-8}$ How do I simplify further? I don't ...
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0answers
25 views

How to make powers work right on a casio calculator?

I pressed some random buttons and it worked fine,but now its not working again. for example my calculator shows that -6^2 is - 36 and not 36, anyone know how to fix it??? My clculator model is casio ...
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3answers
35 views

Derivative of $e^\sqrt{4x+4}$

$$f(x)=e^\sqrt{4x+4}$$ $f(x)=e^u$ $u=\sqrt{4x+4}=(4x+4)^{1/2}$ $u\;'=\dfrac{1}{2}(4x+4)^{-1/2}=\dfrac{1}{2\sqrt{4x+4}}$ I don't know how to proceed from here. Thanks.
2
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3answers
25 views

Calculating sum of consecutive powers of a number

Here is my problem, I want to compute the $$\sum_{i=0}^n P^i : P\in ℤ_{>1}$$ I know I can implement it using an easy recursive function, but since I want to use the formula in a spreadsheet, is ...
0
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1answer
32 views

What is the value of x? Related to Indices.

Just some days ago I appeared for a maths exam. In that exam there was a question related to Indices which I was not able to solve. After the exam I even tried solving it in the home next 2 days but ...
1
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2answers
44 views

Evaluating a limit as $x \to -\infty$ of a power of a rational function

Sorry for the weird title, I don't know how to put the equation on the title. $$\lim_{x\to-\infty}\left(\frac{1-x^3}{x^2+7x}\right)^5$$ Ok I divided inside the parenthesis by $x^2$, but now I am ...
0
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3answers
184 views

Is it true that $0^0$ is undefined? Why or why not? [duplicate]

Is it possible for zero to the power of zero to be undefined? Is there a good reason if it IS undefined? I If yes, I hope there is! This question is different because I'm trying to figure out if 0^0 ...
0
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1answer
47 views

Right way to solve for $\frac{2^{900}*7^{898}}{14^{897}}$

As a sequel to my question How to solve $0.5^{1200}\times (2^{1204})$? : $\frac{2^{900}*7^{898}}{14^{897}}$ Will I first solve the upper raw like did in previous question and then anwser $14^{897}$
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4answers
78 views

How to solve $0.5^{1200}\times (2^{1204})$?

I've been struggling with this one. I know that the anwser is $16$, but how do I solve this on paper? $0.5^{1200}\times 2^{1204}$ I know that this has something to do with first subtracting the ...
-1
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3answers
45 views

Calculating the value with large exponents

I'm trying to solve $2014^{2015}$ $\pmod {11}$, is there a trick or tip to break the problem down to make it easier to solve?
-1
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3answers
39 views

Why is $3^{(x-5)} + 3^{(x-7)} + 3^{(x-9)} = 91$?

So far I think that this is somehow related to that $(x-7) - (x-5) = (x-9) - (x-7) = 2$, but is it ? What steps do you take to add $3^{x-5} + 3^{x-7} + 3^{x-9}$ up ? Thank you!
2
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2answers
31 views

Chaining Exponent Rules Together

I'm having trouble understanding why the following property is true and want to make sense of it before going ahead and use it in my proof by induction: $$2^{2^n}=2^{2^{n-1}}\times 2^{2^{n-1}}=\left( ...
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1answer
27 views

Power function and involution $f(x) = x^a$

For power functions we have a variable $x$ and a constant $a$; we get that $f(x) = x^a$. Find all involutions for $f(x)$. I started out with basic functions such as $f_1(x) = f_1^{-1}(x) = x^1$ and ...
2
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3answers
131 views

Is there a shortcut for raising 2 to the power of a number (e.g. $2^{27}$)?

In networking, when dealing with subnetting, you convert the net mask to binary and count the number of ones (for the example in the question there would be $27$ $1$'s) and to figure out how many ...
1
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3answers
73 views

How can we prove that the square root of any number is equal to the statment given below

Is there any theorem that can explain this $\sqrt[n]{x}= x^{1/n} $, is there any practical example of $ x^{1/n} $, 1/n times of a number.
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0answers
29 views

Relationship of powers of Phi to Lucas Numbers

I was watching a Numberphile and the interviewee was explaining various attributes of Lucas Numbers and he made the statement about creating a sequence by starting with the Golden Ratio and raising it ...
0
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1answer
10 views

Confusion in “mask bits”

This is the question: By definition of the IEEE754 standard, 32-bit floating point numbers are represented as follows: S (1 bit) E (8 bits) M (23 bits) S: Sign bit E: Exponent M: ...
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2answers
27 views

Simplify an expression involving indices

One of my friends asked me this question: Simplify $$\frac{50^{3x-1} 10^{2-3x}}{250^{3x+1}}$$ I've been thinking about the question for more than a day. I've looked through my teacher's notes but ...