Exponent Definition and 212 Threads

Homework Statement Without using a calculator and without evaluating expressions, determine which is greater. 20^100 or 400^40 Homework Equations None, however some exponent laws and possibly mental math are applicable. The Attempt at a Solution I don't know how to reach an answer without...

Homework Statement For which non-zero value of x is the equation -x^ -4 = (-x)^ -4 true? Explain. Homework Equations None. Other than applicable exponent laws. The Attempt at a Solution I know how to use the guess and check method. But I was wondering how to reach the answer faster and how...

Initially the expression A has 8 terms. So how is it reduced in the second line to 5 terms? Could you show me, please? Thank you. \begin{align*} A&=10^{28} -10^{22} +61\times10^{14}+12\times10^{21}-12\times10^{15}+3\times10^{9}-36\times10^{8}+9\times10^{2}\\ &=10^{28}...

First it starts as r= p* (50K^-.5 100^.5) then K=[(50p100^.5/r]^2 So how does the power of 2 get there in the second part when moving K to the other side?

In computational physics is very often to calculate largest Lyapunov exponent. If largest Lyapunov exponent ##LE## is positive there is chaos in the system, if it is negative or zero there is no chaos in the system. But what can we say about some certain value of ##LE##. For example...

Homework Statement I. 3*3 matrix A (8 2 -2, 2 5 4, -2 4 5) II. 3*3 matrix (1 2 0, -1 -2 0, 3 5 1) Homework Equations I. Solve Aexp 100 of 3*3 II. Find the 5th rooth of B matrix The Attempt at a Solution I. I got stuck at diagonalising the matrix. Is this OK 1st step ? If yes...

So, essentially, all I wonder is: What is the The Matrix Exponent of the Identity Matrix, I? Silly question perhaps, but here follows my problem. Per definition, the Matrix Exponent of the matrix A is, e^{A} = I + A + \frac{A^2}{2} + \ldots = I + \sum_{k=1}^{\infty} \frac{A^k}{k!} =...

Homework Statement log39x4 - log3(3x)2 The answer sheet says that the answer is 0, but when I work the problem both ways I get: 2log3x Any ideas? Homework Equations logxy/z = logxy - logxz The Attempt at a Solution Formula Sheet 20-30 Minutes of Messing around with the problem

My TI-89 calculator rearranged $$ 2^{2m+1}-1$$ to $$ 2\times 4^m - 1 $$ I can't for life of me work out how it did it. Is anybody able to put me out of my misery? Thank you in advance! Ben

I'm using Pascal's (n choose k) method for calculating the coefficients of the terms of a binomial expansion. However, if the exponent is a negative integer, how can one use this method, seeing as factorials for negative integers are undefined. For example, how could one determine the...

Homework Statement In writing the definition of ##e## i.e. ##e=\displaystyle\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n##, why do we denote the variable by 'n' despite the fact that the formula holds for n∈(-∞,∞)? Is there any specific reason behind this notation i.e. does the variable have...

Why is base to power 0 always 1, even if it's .2^0 it =1? Is it just counting the 1 time the .2 is existing? Is that why?And - base ^ 0 is negative whatever the number is so why is that not a negative one? Because (-2)^2 is 4 but no because a negative times a negative is a positive. And...

For some metals and alloys the region of the true stress–strain curve from the onset of plastic deformation to the point at which necking begins may be approximated by σ=Kεn where n is strain hardening exponent. I wonder whether this equation can be applied to engineering stress-strain curve or...

The problem is to verify ##(g^n)^{-1} = g^{-n}## is true ##\forall n \in \mathbb{Z}##. Here is my proof: ## (g^n)^{-1} = (\underbrace{g \star g~ \star ~...~ \star g}_{n~ \mbox{copies}})^{-1} \iff## ##(g^n)^{-1} = [(g \star ~...~ \star g) \star g]^{-1}## Using ##(a \star b)^{-1} = b^{-1}...

How do you solve when an exponent is pi? And a cube root. Thanks, sorry I'm slow.

hello! I want to know what happens, what means, what the properties are etc of a negative exponent I read on wikipedia "When n is a negative integer and b is non-zero, b^n is naturally defined as 1/b^−n" So based on the above, 3^-2 = 1/3^2 1) Is this correct? 2) Why does this...

Something i ran into while doing hw Homework Statement starting with \int{dx} e^{-ikx}\delta(x) = 1 we conclude by Fourier theory that \int{dk} e^{+ikx} = \delta(x) Now, i try to compute \int{dk} e^{-ikx} (I've dropped the normalization factors of 2\pi. I believe no harm is done by...

Is there a good general definition of 0^z, where z may be complex? The cases where z is real (and positive, negative, or zero) are straightforward, but what if z isn't real? Are there arbitrary branch cuts involved, or is there some universal definition?

How come when you do something like -3^2=-9. Does it mean that the dash is a subtraction symbol and not a negative symbol?

Problem: Let $y=x/(1+x)$, where $$\Large x=\omega^{2009^{2009^{\cdots \text{upto 2009 times}}}}$$ and $\omega$ is a complex root of 1. Then $y$ is A)$\omega$ B)$-\omega$ C)$\omega^2$ D)$-\omega^2$ Attempt: I somehow need to show that the huge exponent is of the form $3k$, $3k+1$ or $3k-1$...

My confusion comes from basic exponent rules and whether or not both sides of an equation have to have the same level of exponent, when you reduce the base for solving. If one side can have an exponent of 3, does the other side also have to be reduced to something that would also have an...

Homework Statement Is it true that: (ABA^-1)^8 = AB^8A^-1 for all n x n matrix and not just for invertible matrix? My attempt: (ABA^-1)^8 =A^8B^8A^-8

Hey I didn't understand the transition below, I'd be glad for some help thanks

Closed form for "geometricish" series (index squared in the exponent)? Hi all, Is there a nice closed form for the following series? \sum_{k=0}^n x^{k^2} Even a decently tight upper bound and lower bound would be nice (obviously it is bounded by the corresponding geometric series \sum...

Hi MHB, Problem: Solve in the set of real numbers the equation $5^x+5^{x^2}=4^x+6^{x^2}$. Attempt: At first glance, we can tell $x=0, 1$ would be the two answers to the problem but how do we prove these two are the only answers? I think this problem must have something to do with the Mean...

problem is solved.

I'm trying to manipulate (x+1)^x+1 / ((x+1)+1)^x+1 So that I have a 1 in the numerator. If I bring the numerator down using the integer exponent rule, I'll have... 1 / ( (x+1) / (x+1) + 1 )^x+1 ? Whoops, that's not right... 1 / (x+1)^x * ((x+1)+1)^x+1 ?

These are the two last problems I'll bother you with for a short while (I love this forum, I'll definitely stay on and hopefully be able to contribute in the future). Homework Statement Problem 1: (##-x^2##-1)sin2x > 0 , xe[0,2\pi] Problem 2: ##2^{-x^2+x+2}## < 4 Homework...

Prove that $$\left( 6+845^{\frac{1}{3}}+325^{\frac{1}{3}} \right)^{\frac{1}{3}}+\left( 6+847^{\frac{1}{3}}+539^{\frac{1}{3}} \right)^{\frac{1}{3}}=\left( 4+245^{\frac{1}{3}}+175^{\frac{1}{3}} \right)^{\frac{1}{3}}+\left( 8+1859^{\frac{1}{3}}+1573^{\frac{1}{3}} \right)^{\frac{1}{3}}$$

Homework Statement So, after performing an experiment a number of times, to calculate the efficiency of a singular repeating compressor, I have found n ≈ 1 i.e. it equals 1.01... in all of my found values. I've now been asked to classify the type of process. Seeing as technically n > 0 and n<...

I need to find the derivative of R(t)=5t-3/5 are there any derivative rules I can use for this problem?

Solve the DE by using separation of variables \frac{dy}{dx} = e^{3x+2y} Break up e^{3x+2y} = e^{3x}e^{2y} Move x's and y's to their own side of the equation. \frac{1}{e^{2y}} dy = e^{3x} dx Integrate both sides of the equation to get \frac{-e^{2y}}{2x}=\frac{e^{3x}}{3}+C I don't know how to...

I'm curious but usually negative exponents mean you're going to put those numbers on the bottom (sorry forgot the exact term it is much earlier than I normally get up) (example 6^-1 =1/6 so why is scientific notation different? Or is it? (Ex: 5x10^-3=.005)

I have been out of school for 35 years and this is out of my reach. Can someone please re-arrange this formula solved for Tp . T = (Tp / Sa) to the .833 power I have an example in it's current form. T = 40.5 Deg C Tp = 2300mW/CM cubbed Sa = 27 CM squared

Homework Statement Find the derivative of (sin x) ^ ((sin(sin x)))Homework Equations The Attempt at a Solution I get sin(sin x) * [(sin x) ^ {(sin(sinx)-1)}* cos x] The cos x isn't part of the exponent Is this right? Thanks :)

When proving that x^m x^n = x^{m+n} and that (x^m)^n = x^{mn} for all elements x in a group, it's easy enough to show that they hold for all m \in \mathbb{Z} and for all n \in \mathbb{N} using induction on n. The case n = 0 is also very easy. But how does one prove this for n \in...

Homework Statement Calculate the Lyapunov exponent for the linear map xn+1= rxn. Homework Equations λ = Lyapunov Exponent λ = \lim_{n \rightarrow \infty} \begin{bmatrix}\dfrac{1}{n} \sum_{i = 0}^{n - 1} ln|f'(x_i)| \end{bmatrix} The Attempt at a Solution f'(x) = r. λ =...

Homework Statement This isn't really a specific problem, just a question if hand-writing log functions (or trig functions) is interpreted differently than when typing them into a calculator or something like Wolfram Alpha. Suppose you have this on paper: ln ex Is this the same as...

I have the following equation that I'm trying to simplify: $$\frac{5 + \sqrt{5}}{2\sqrt{5}}*(\frac{1 + \sqrt{5}}{2})^{x} $$ From looking at it, it seems like it could be simplified so that the right-hand side of the multiplication would be: $$(\frac{1 + \sqrt{5}}{2})^{x+1} $$ I started to...

Homework Statement x^X^x^x^x=2. find value of x. Homework Equations taking log both sides, but it makes a equation which i am not able to solve. The Attempt at a Solution x^(x)^4=2 x^4logx=log2. what next?

Hey, it is clear for me that \sum_{i=1}^{n} i = \frac{n(n+1)}{2} how to find a formula for \sum_{i=1}^{n} i^2 \sum_{i=1}^{n} i^3 Thanks

a^x means a*a*a*a... x times This makes sense for whole numbers to me, but I am sort of lost about transfering the definition one x becomes a fraction. I know that the denominator in a fraction in an exponent means "the denominator root"..., and I know what a root its. But it is a jump in...

Homework Statement So I had to find out N=Noe-ux Where No=1.5,e is the elementary charge, u=-0.068 and x=0.07 I came to the answer 1.221 Now I need to find x if u =-0.036, N=1.221 and No remains 1.5Homework Equations The Attempt at a Solution I reduce the equation to the following...

Homework Statement ([2x+1/4x+3]^2) Homework Equations Exponent and quotient rule The Attempt at a Solution Would this become: 2* (2x+1/4x+3) then do the quotient rule?

Def. Let $\{z_j\}$ be a sequence of non-zero complex numbers. We call the exponent of convergence of the sequence the positive number $b$, if it exists, $$b=inf\{\rho >0 :\sum_{j=1}^{+\infty}\frac{1}{|z_j|^{\rho}}<\infty \}$$ Now consider the function $$f(z)=e^{e^z}-1$$ Find the zeros $\{z_j\}$...

$\sum\limits_{n = 2}^{\infty}n^p\left(\frac{1}{\sqrt{n - 1}} - \frac{1}{\sqrt{n}}\right)$ where p is any fixed real number. If this was just the telescoping series or the p-series, this wouldn't be a problem.

Homework Statement Consider the matrix cos(3*pi/17) -sin(3*pi/17) S = sin(3*pi/17) cos(3*pi/17) Does there exist a positive integer n such that Sn = I where I is the 2x2 identity? If so, what is the smallest such integer? Explain. Excuse the poor matrix...

I have a theory that i need to prove but I am not quite sure how to mathematically prove that it is true. Theory: When you square a rational number, each of the prime factors has an even exponent. For example, 10 --> If i square 10, which is a rational number, =10^2 =(5^2 x 2^2)...

Homework Statement f(x)=x^7/2, Find the derivative Homework Equations f'(a)=limit as x approaches a; ((x+h)-x)/h The Attempt at a Solution limit as h approached 0; ((x+h)^7/2-x^7/2)/h Multiplied by the conjugate with -5/2 as exponent and came up with 2x^5/2, which is incorrect

On a scientific calculator, how do you get a root of a number for an exponiental? Ex.-.653846^1/12= I was taking 1/12 and making it so I can make it exponent for .653846. Obviously wrong. What am I doing here??