Questions about exponentiation

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0answers
19 views

Logically these are not equivalent

My electronics professor told me that removing the $T^3$ from equation (A) makes a minimal difference between (A) and (B) when solving for $T$. It clearly does not. What is the best way to go about ...
1
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0answers
44 views

How to prove that $b^{x+y} = b^x b^y$ using this approach?

Fix $b>1$. If $m$, $n$, $p$, $q$ are integers, $n > 0$, $q > 0$, $r = m/n = p/q$, then I can prove that $(b^m)^{1/n} = (b^p)^{1/q}$. Hence it makes sense to define $b^r = (b^m)^{1/n}$. I ...
3
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2answers
304 views

Algorithm for computing powers

I was challenged by one of my fellow students to write a mini-library in the programming language called C that enables you to work with very large numbers (the numbers that the language offers ...
0
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1answer
29 views

From an expression raised in a power of 2 to an expression raised in the power or 10

Is there a simple/"easy" way to convert a big number from a power of $2$ to a power of $10$ equivalent. Example: I had $2^{127}\cdot 1.9999999$ which I did the multiplication got the result and from ...
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4answers
40 views

Why is a Constant added to front?

I made the differential equation : $$dQ = (-1/100)2Q dt$$ I separate it and get: $\int_a^b x (dQ/Q) = \int_a^b x (-2/100)dt$ this leads me to: $\log(|Q|) = (-t/50) + C$ I simplify that to $Q = ...
0
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1answer
35 views

Big-Oh of exponent of exponent

How does one whether an exponent of an exponent is the big-Oh of the other? For example, if I have $a^{b^n}$ and $b^{a^n}$, how would i determine and prove which is a big oh of another? I'm thinking ...
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1answer
52 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 ...
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1answer
24 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 $4^5$
3
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0answers
23 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 ...
-1
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2answers
25 views

Exponentiation expression [closed]

This is my first participation, seeking the solution for the following expression: Any help is appreciated, with thanks in anticipation. Regards.
3
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2answers
90 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
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1answer
78 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 $x$ ...
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4answers
58 views

Explain the answer of $0.0625^{-2.25}$ [closed]

Explain the answer of $0.0625^{-2.25}$ I know the answer but how it will be?
0
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1answer
19 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
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2answers
52 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
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1answer
34 views

exponent y=x^a sequences.

While analyzing square and cube functions, i found the following: ...
6
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4answers
305 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 ...
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1answer
58 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?
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0answers
27 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 $\exp(aD)f(x)=f(x+a)$?
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3answers
39 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 ...
12
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3answers
317 views

Calculate $\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$

I'm an eight-grader and I need help to answer this math problem. Problem: Calculate $$\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$$ This one is very hard for me. It ...
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3answers
48 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 ...
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3answers
89 views

Trouble Understanding Large Number [closed]

Alright so I want to know how large this number actually is. I understand that ten to the power of ten is ten billion. What do you do when there is a power and then another power? How large is the ...
12
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1answer
301 views

If $n^c\in\mathbb N$ for every $n\in\mathbb N$, then $c$ is a non-negative integer?

Supposing that a real number $c$ is given, is the following true? "If $n^c$ is a natural number for every natural number $n$, then $c$ is a non-negative integer." Though this seems true, I can't ...
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1answer
62 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?
1
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1answer
94 views

Generalisation of Lambert W function?

I want to solve an equation of the form: $\exp(C / x) - 1 = D / (x + a)$ This seems to be almost in a form where I can express solutions in terms of the Lambert W function but I can't seem to figure ...
3
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4answers
92 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 ...
0
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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 ...
2
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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 ...
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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
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6answers
78 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 $?
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3answers
121 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
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2answers
29 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
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5answers
106 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?
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3answers
60 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
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1answer
89 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
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2answers
59 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
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1answer
71 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
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1answer
93 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
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2answers
48 views

Matrix Exponent - equivalent of a rotation matrix

For every Rotation Matrix,there is a Matrix Exponent representation where the power is a skew symmetric matrix. More clearly if I have a rotation matrix ${R}_{3 \times 3}$ then there will be a skew ...
4
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3answers
3k views

$x^x=y$. How to solve for $x$?

I tried looking for ways to solve this equation and came across something like Lambert's W function, which, by the way, I did not understand a bit, because I've never learned it nor do I have a decent ...
5
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3answers
121 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 ...
6
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2answers
220 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
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2answers
44 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 ...
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3answers
90 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} ...
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1answer
30 views

Using exponent laws to simplify $2^t\times4^2+8^{t-2}$

This is one of the questions in my maths text book and I don't quite get it, can someone please explain it and show the working out? use index laws to find $2^t\times4^2+8^{t-2}$ Observation ...
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3answers
67 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 ...
4
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3answers
119 views

Solve for $x$ in the equation [closed]

Please help me to solve for x using maybe logarithm or exponential rules (or both) $$ 5^x=2 \cdot 3^x $$
2
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2answers
60 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
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4answers
453 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.