Tagged Questions

Questions about exponentiation

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0
votes
1answer
19 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.
1
vote
0answers
19 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
votes
1answer
15 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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0answers
37 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
votes
1answer
47 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
votes
3answers
77 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
votes
1answer
46 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
vote
2answers
41 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
46 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
vote
0answers
59 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
votes
2answers
22 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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votes
3answers
104 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?
0
votes
3answers
40 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
votes
0answers
27 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
votes
1answer
44 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 ...
0
votes
0answers
25 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 ...
1
vote
2answers
31 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
votes
0answers
25 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
vote
3answers
18 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
votes
2answers
48 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) ...
0
votes
2answers
35 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
21 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 ...
1
vote
3answers
30 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
votes
1answer
31 views

The sum of powers of two [on hold]

While doing a proof of correctness for an algorithm, I ran into a roadblock with a smaller proof. The problem in words: A set with all elements which are a power of 2, and the sum of the set is ...
2
votes
3answers
22 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
votes
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 ...
0
votes
2answers
40 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 ...
-1
votes
3answers
144 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? If yes, I hope there is! This question is different because I'm trying to figure out if $0^0$ ...
0
votes
1answer
46 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 $14^{897}$
3
votes
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
votes
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
votes
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
votes
2answers
27 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( ...
0
votes
1answer
23 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
votes
3answers
100 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
vote
3answers
62 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.
0
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0answers
17 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
votes
1answer
9 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: ...
0
votes
2answers
26 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 ...
1
vote
2answers
44 views

Solving equations with fractional indices

How would I go about solving an equation like this? $3^{4/3}b^{5/3} - b^3 = 1$ I thought about rearranging to get $3^{4/5}b = (1 + b^3)^{3/5}$, but that didn't seem to lead anywhere as I couldn't ...
0
votes
2answers
17 views

Exponents with the power being a negative/decimal?

How would I do the equation $b^{n}$ Or $b^{-n}$ Where $b$ is the base, and $n$ in a negative/decimal?
16
votes
6answers
2k views

What's the inverse operation of exponents?

You know, like addition is the inverse operation of subtraction, vice versa, multiplication is the inverse of division, vice versa , square is the inverse of square root, vice versa. What's the ...
0
votes
1answer
41 views

Irrational power of root

Let $a$ and $b$ be rational numbers, such that $\sqrt{a}$ and $\sqrt{b}$ are irrational. Can $\sqrt{a}^\sqrt{b}$ be rational? I found examples, where the irrational power of an irrational number is ...
0
votes
1answer
72 views

Simplifying $2^\sqrt{\log x}$

Can this expression be simplified? $$2^\sqrt{\log x}$$ Thank you
1
vote
2answers
38 views

Power of very big numbers

So, I was solving a question, and I came across this. If I have, x=a^b, and If I want to calculate the last digit of x, then it ...
1
vote
4answers
77 views

$4^{x}+2^{x+1}=18$ Please help me solve?

I tried using logs on both sides or tried treating it as a quadratic but didn't manage to simplify it, Help?:D
0
votes
1answer
15 views

exponential equation (solve for x)

We started review in my calculus class and I have mostly forgotten everything about exponential equations. $$ e^{-x} * (3e^{2x}-(5/4))^{1/2} = e^x $$ Would x just equal 0, when -x+x=0 (two sides of ...
1
vote
1answer
64 views

is there any solution for $x^2 +x + 2 = e^x$ by using algebra?

I know this can be solved by numerical methods but I would like to know whether this can be solved using logs or something similar. Thanks
2
votes
3answers
125 views

Matrix exponential: Formal notation for power series? Or, more?

For a square matrix $A$, I'm already used to see and use: $$\sum_{n=0}^{\infty} \frac{A^n}{n!} = \lim_{n \to \infty} \left(I + \frac{A}{n}\right)^n = e^A$$ Which means a matrix $A$ is just like some ...
1
vote
2answers
29 views

Negative Number raised to fractional power

How would you solve a negative number raised to a fraction a/b if b is odd and a is evem? Ignoring imaginary numbers i.e $(-1)^\frac23$ Calculator returns an error $(-1)^\frac 13 (-1)^\frac 13$ = ...