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

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3
votes
4answers
73 views

Existence of solution in $x,y \in (a,b)$ of $ \bigg(\dfrac { a+b}2\bigg)^{x+y}=a^xb^y$

Let $a<b$ be positive real numbers , then is it true that there exist $x,y \in (a,b)$ such that $ \bigg(\dfrac { a+b}2\bigg)^{x+y}=a^xb^y$ ?
0
votes
1answer
34 views

Methods to calculate powers in my head

How can i calculate powers in my head, like small powers. for example $0.5^3$, how can i work this out quickly and easily? or $4^5$
1
vote
1answer
55 views

Definition of $z^0, z=a+bi, a,b \in \mathbb{R}, z \neq 0$

Today I found in a True-False the question; Does the equality $$z^0=1, z=a+bi, a,b \in \mathbb{R}$$ hold $\forall z \in \mathbb{C^*}$? The thing is, this was never clearly defined in the book, and ...
3
votes
0answers
25 views

Iterations $n, n^n, (n^n)^{(n^n)},…$

(Note: I'm reposting this, as I posted the original too late in the evening to gain anyone's notice.) A contest problem (#2 on the 2010 Virginia Tech Math Competition) proffers the solver the ...
4
votes
2answers
97 views

Solve the equation: $1+2^x+4^x+8^x+16^x+32^x=3(1+2^x+4^x)$

I am doing some math repetition and am a bit stuck on this exercise: Solve the equation: $1+2^x+4^x+8^x+16^x+32^x=3(1+2^x+4^x)$. Now, this is a geometric sum on both the $LHS$ and $RHS$, which I ...
1
vote
1answer
90 views

Can matrix exponentials ever be negative? If so, under what conditions?

Let $C$ be a $2 \times 2$ matrix with real entries, and $x\in\mathbb{R}^2$. We write $x > 0$ if both coordinates are strictly positive. Suppose $x>0$, under what conditions on $C$ and $x$ ...
-6
votes
4answers
109 views

Computing $0.0625^{-2.25}$ without a calculator

It is quite easy to see that $0.0625^{-2.25} = 512$ by plugging this into a calculator. Of course, mathematics existed for millennia before the invention of the calculator; is there a way to compute ...
0
votes
1answer
21 views

Complete the square for exponents

How does exp(2x)-2+exp(-2x) = (exp(x) - exp(-x))^2 I am having trouble using complete the square
-1
votes
2answers
57 views

If a matrix is non diagonalizable, what other method can I use to calculate the nth power?

First off, I have this matrix A: 1 0 3 1 0 2 0 5 0 I have calculated the eigenvalues, which are ...
1
vote
1answer
34 views

exponent y=x^a sequences.

While analyzing square and cube functions, i found the following: ...
0
votes
1answer
61 views

How does one obtain the expansion of $e^{-x^2}$ in a power series?

So I know that the Power Series $y = \displaystyle\sum_{m=0}^\infty\displaystyle\frac{(-1)^m}{m!} x^{2m}$ is equivalent to $e^{-x^2}$. Could someone show me why this is?
1
vote
0answers
36 views

Exponential of a power of the differential operator

In relation to this question: Exponential of a polynomial of the differential operator Is there an expression for $\exp(aD^n)f(x)$ similar to $\exp(aD)f(x)=f(x+a)$?
1
vote
3answers
42 views

Easy exponentiation method

Is there a simple way of solving, say, x^(3/2)? For example, one way of solving 16^(3/2) is to calculate the square root of 16^3, but I was wondering if there is a simpler mental trick for doing this ...
1
vote
3answers
92 views

How to calculate power of a number with decimal exponent programatically?

I am trying to code and algorithm that can allow me to calculate power of a function with decimal exponent. The language that I am using to code in doesn't has any predefined power functions. I ...
3
votes
4answers
96 views

If $q^n$ is irrational for all $n>1$, then $q$ is irrational.

Theorem. Let $q \in \mathbb{R}$ an arbitrary given number. If $q^n$ is irrational for all $n>1$ integer, then $q$ is irrational. My Questions. What is a the name of this statement and what is the ...
1
vote
1answer
63 views

Showing that $\exp(\sum_{n=1}^\infty a_nX^n)=\prod_{n=1}^\infty\exp(a_nX^n)$ for formal power series

I've just come across formal power series and am not very fluent with them yet. I'd like to show that $\exp(\sum_{n=1}^\infty a_nX^n)=\prod_{n=1}^\infty\exp(a_nX^n)$. Can anybody help?
0
votes
2answers
28 views

Why is this term $=1$

Can you tell me why $$\frac{1}{r} \sum_{k=0}^{r-1} R_N(x^k) \sum_{s=0}^{r-1} e^{\frac{-2 \pi i s k}{r}}=1?$$ Here $R_N(x^k)$ is the remainder of $x^k$ Modulo $N$. When I entered the last sum in ...
0
votes
1answer
36 views

How to divide $6^{4/6}$ by $6^{6/8}$?

How do you resolve this: base 6 and exponent 4/6 divided by base 6 and exponent 6/8?
2
votes
6answers
81 views

Why is the result of $-2^2 = -4$ but $(-2)^2 =4$?

I am really new into math, why is $-2^2 = -4 $ and $(-2)^2 = 4 $?
0
votes
2answers
32 views

Distributing exponents to variables [duplicate]

I am studying parabolas, but the way exponents are distributed is confusing me to oblivion. $(y_2 – 2)^2$ $y_2^2 – 4y_2^2 + 4$ I do not understand how can the first expression be simplified to the ...
2
votes
3answers
48 views

How to multiply the binomials $(2x^3 - x)\left(\sqrt{x} + \frac{2}{x}\right)$

I am sorry if the numbers are not formatted, I have searched but found nothing on how. I am trying to multiply $$(2x^3 - x)\left(\sqrt{x} + \frac {2}{x}\right)$$ together and I arrive at a different ...
2
votes
5answers
108 views

How does $2^n + 2^n = 2^{n+1}$?

What property of exponents can be used to show that $$2^n + 2^n = 2^{n+1}$$ Does this work for all constants raised to a variable exponent?
2
votes
4answers
164 views

How is $ i^{-1} = -i$ and $i^{-3} = i$?

Now I know that with positive powers of $i$ the cycle is: $i , -1 , -i , 1\ldots$ The negative power cycle is: $-i , -1 , i , 1 \ldots$ Can someone explain to me how $\frac 1 {\sqrt{-1}}$ is equal ...
0
votes
3answers
61 views

modules with several powers $x^{y^z}$

I have an assignment(which gives no credit) which I cant solve so I would like to get some help $3^{{1001}^{1002}}\pmod {43}$ (All original numbers were replaced) Our hint was to use the ...
1
vote
1answer
91 views

Inequality: $\left|x^3-y^3\right|<|x|^3+|y|^3$

Could anyone show me why $$\left|x^3-y^3\right|<|x|^3+|y|^3$$ for all real numbers (x,y) except 0? I'm thinking of whether of how to remove the modulus sign on the left hand side of the ...
1
vote
2answers
60 views

How does exponentiation relate to multiplication?

My book derives the logarithm function as a definite integral of $1/x$ and defines the exponential function as its inverse. It then extends this definition to other bases: $$b^x = e^{\ln (b) x}$$ ...
3
votes
1answer
72 views

Integral of an exponent of an exponent

For a homework problem, I have to integrate this: $$\int{4^{(4+x)^x}}dx$$ How would I go around to starting this question? I don't know how to evaluate this, and I've tried to use u-subs and ...
3
votes
1answer
95 views

How can I do this? $\int\frac{dx}{x^4+1}$ [duplicate]

I tried to integrate this: $\displaystyle\int \dfrac{dx}{x^4+1}$ I tried to do it with the partial fractions method (after factoring the denominator), but the process is really large, and I got a lot ...
0
votes
2answers
63 views

Matrix Exponent - equivalent of a rotation matrix

Every Rotation Matrixcan be represented as a power of e with exponent a skew symmetric matrix. In particular, if we have a rotation matrix ${R}\in\mathbb R^{3 \times 3,}$ then there will be a skew ...
6
votes
2answers
223 views

Evaluating a limit. What makes the equality right?

I'm reading a proof of a limit calculation. The limit is: $$\lim\limits_{x\to 0}\left(\frac{a^x+b^x}{2}\right)^\frac{1}{x}$$ where $a,b>0$. The aother claims that: $$\lim\limits_{x\to ...
0
votes
2answers
46 views

When $\ln(1+y) = y + o(y)$?

I was reading a proof which utilize the fact that: $\ln(1+y) = y + o(y)$ http://math.stackexchange.com/a/842557/160028 I'm not so sure what is the meaning of $\ln(1+y) = y + o(y)$. When is it ...
1
vote
3answers
91 views

Does $x^2+ x^4/(3\cdot4) + x^6/(3\cdot4\cdot5\cdot6) + \cdots$ have any compact form?

Is there any compact form for the following series $$F_1(x) = x^2+ \frac{x^4}{3\cdot4} + \frac{x^6}{3\cdot4\cdot5\cdot6} + \cdots$$ $$F_2(x) = x+ \frac{x^3}{2\cdot3} + \frac{x^5}{2\cdot3\cdot4\cdot5} ...
1
vote
3answers
68 views

generalized way of finding pair solutions of an equation

I want to find out pair solutions of this equation: $$x^{2}-79y^{2}=1$$ This is a hyperbola equation. I sketched its graph, but that didn't help me. I think the square from (form?) of $x$ and $y$ is ...
2
votes
2answers
62 views

Evaluate $\frac{\frac{1}{1}}{\frac{1}{5^{-2}}}$

I solved a question in the Manhattan GRE 5 pound book (specifically the 11th question in the Exponents and Roots section). I evaluated $\frac{\frac{1}{1}}{\frac{1}{5^{-2}}}$ as $5^{-2}$ and then ...
2
votes
4answers
458 views

Solving an equation in $x$, in which $x$ occurs as exponent four times

Find the number of solutions to the equation $$2011^x+2012^x+2013^x-2014^x=0$$ The answer seems to be zero, but I have no idea why. Please avoid considering complex solutions and other scary things.
3
votes
2answers
75 views

Solution for this Logarithmic Equation

Recently I was going through a problem from the book Problems in Mathematics - *V Govorov & P Dybov* . $$(x-2)^{\log^2(x-2)+\log(x-2)^5-12}=10^2\log(x-2)$$ I tried solving by first considering ...
2
votes
2answers
77 views

How can I calculate the median value?

$$f(a,b) = a^b$$ Where $0\le a \le1$ and $0\le b \le1$ and either $a\ne0$ or $b\ne0$ How can I calculate the median value of $f$ ? I can estimate it to be about 0.76536 by taking values along the ...
5
votes
3answers
126 views

Prove $\lim_{n\to\infty} \frac{(2^{2^n}+1)(2^{2^n}+3)(2^{2^n}+5)\cdots (2^{2^n+1}+1)}{(2^{2^n})(2^{2^n}+2)(2^{2^n}+4)\cdots (2^{2^n+1})}=\sqrt{2}$

Question: Prove or disprove $$I=\lim_{n\to\infty} \frac{(2^{2^n}+1)(2^{2^n}+3)(2^{2^n}+5)\cdots (2^{2^n+1}+1)}{(2^{2^n})(2^{2^n}+2)(2^{2^n}+4)\cdots (2^{2^n+1})}=\sqrt{2}$$ I know ...
4
votes
0answers
70 views

Definition of $0^z$, $z$ complex

Usually we define the function $a^b$ over the complexes using the exponential function, as $e^{b\log a}$. This function has some issues with multivalued-ness, but it still more or less satisfies ...
3
votes
4answers
117 views

Solving $e^{4x}+3e^{2x}-28=0$

How to solve this equation: $$e^{4x}+3e^{2x}-28=0$$ I don't know how to solve this problem. I read over another example, $e^{2x}-2e^x-8=0,$ and it said that $e^{2x}$ is $e$ to the $x$ squared, ...
0
votes
3answers
49 views

Converting exponents to scientific notation

I have to solve or estimate the answer to an equation that is as follows: $$P_\text{blocks} = \frac{398 \cdot 19^{65}}{\prod^{66}_{i=0} 78804 - i}$$ It doesn't take long to realize that this is an ...
1
vote
1answer
38 views

Transforming a power tower to a product

It is possible to write the product of a sequence of terms $a_i$ as a function of the sum of a sequence of functions of these terms: $$\prod_i a_i=f\left(\sum_i g(a_i)\right)$$ where $f=\exp$ and ...
2
votes
1answer
49 views

Convergence of an infinite power

There are complex numbers $z$ and $w$ for which $$\lim_{n\rightarrow\infty}z\uparrow\uparrow n=w$$ where $\uparrow\uparrow$ is the tetration symbol, e.g. $z=\sqrt{2}$ and $w=2$. Are there complex ...
-3
votes
2answers
71 views

How can I simplify the expression $\frac{\sqrt[5]{x^2}}{x^2}$?

$$\dfrac{\sqrt[5]{x^2}}{x^2}$$ I'm doing a summer math packet for calculus. I need to simplify the above. I think I may know the answer, but I'm not sure. Can someone help me, please?
3
votes
3answers
71 views

$x^y < y^x$ for $y\ll x$?

Sorry if this is a naive question; I am not very good at mathematics. It seems obvious that for many $x$ and $y$, $x^y < y^x$ if $y \ll x$, e.g. $2^{10} > 10^2$. If $x$ and $y$ are very close ...
0
votes
1answer
43 views

Why, or why not, is $5^{log_3(n)} \in \mathcal{O}(n^2)$?

Why, or why not, is $5^{\log_3(n)} \in \mathcal{O}(n^2)$ ? I tried transforming the logarithm to a base of 5, so that the logarithm and power cancel each other out. However, when I try to so I get ...
2
votes
1answer
116 views

Combination of quadratic and cubic series

I'm an eight-grader and I need help to answer this math problem (homework). Problem: Calculate $$\frac{1^2+2^2+3^2+4^2+...+1000^2}{1^3+2^3+3^3+4^3+...+1000^3}$$ Attempt: I know how to ...
1
vote
1answer
69 views

Square root of $\frac{2}{2^x}$; how do I find $x$?

I have this: $$\sqrt{\frac{2}{2^x}} = 9.313225746154785 \times 10^{−10}$$ (sqrt(2/(2^x))) How should I find $x$? I know it's 61 for this case, but I'd like to know how to solve it for when I don't. ...
6
votes
2answers
160 views

Calculating limit in parts. Why possible?

Let $f$, continuous function, differentiable at $x=1$ and $f(1)>0$. Consider the following equation: $$\lim \limits_{x\to 1} ...
1
vote
1answer
48 views

How to show that $w$ is a $N$th primitive root of unity?

I am studying the discrete Fourier transform. For sequence $x_{0}, \dots, x_{N-1}$ it is defined as $$X_{k} = \sum_{n=0}^{N-1} x_{n}e^{-2\pi ikn/N} \quad 0 \leq k \leq N-1$$ according to Wikipedia. ...