Questions about exponentiation

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2answers
24 views

Non-integer exponents of negative numbers?

There is a formula for exponents of negative numbers as follows: $m^n=(-1)^n|m|^n$. This formulation works when $m<0$ and $n\in \mathbb{Z}$. But what about for $n\in \mathbb{R}$? Is there a ...
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2answers
24 views

no. and nature of roots of $x^{\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}} = \sqrt{2}$

The given equation is $$x^{\frac{3}{4}(\log_{2}{x})^2 + \log_{2}{x} - \frac{5}{4}} = \sqrt{2}$$ I took $\log_{2}{x}$ = $t$ and then rewrote the given equation as $$x^{3t^2 + 4t - 5} = \sqrt{2}$$ ...
3
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3answers
229 views

Which is greater: $1000^{1000}$ or $1001^{999}$

Question: Find the greater number: $1000^{1000}$ or $1001^{999}$ My Attempt: I know that: $(a+b)^n \geq a^n + a^{n-1}bn$. Thus, $(1+999)^{1000} \geq 999001$ And $(1+1000)^{999} \geq ...
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1answer
47 views

Exponential problems

A ship embarked on a long voyage. At the start of the voyage, there were 300 ants in the cargo hold of the ship. One week into the voyage, there were 600 ants. Suppose the population of ants is an ...
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2answers
46 views

No real solution to logarithmic equation?

$$e^x + 1 = 2e^{-x}$$ Wolfram Alpha claims no real solution and my text book claims the solution $x=0$. Why can't I simply multiply each side by $e^x$: $$e^{2x} + e^x = 2$$ $$\ln(e^{2x}) + ...
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1answer
41 views

If $\log_{30}{3} = c$ and $\log_{30}{5} = d$ then the value of $\log_{30}{8} $ is??

I attempted the following: $\log_{30}{8} = 3\log_{30}{2}$ $\log_{30}{3} = c$ is equivalent to $3 = 30^c$ $\log_{30}{5} = d$ is equivalent to $5 = 30^d$ What should I do further?? Is ...
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1answer
10 views

Multiplying radicals expressing a single radical

I am having problems understanding how to multiply radicals i.e (($√5 )(^3√2)$). I know the answer is $ ^6√500 $. I just do not understand how to get there. Update: I've solidified my understanding ...
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3answers
42 views

Fractional Exponents powers

I am having problems understanding how to answer questions containing fractional exponents to a given power ie $(2x^{1/2})^6$, i do not understand how to go about answering the question. I know this ...
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2answers
118 views

When is the power of a binomial equal to the sum of like powers of its terms?

Question: Under what circumstances/restrictions on $x$ and $y$ does $(x + y)^n = x^n + y^n$ given the value of $n$? That is, what can we tell about $x$ and $y$ from the value of $n$ and the equation ...
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2answers
26 views

Exponential Growth Rates

So if you are given two different numbers to determine a growth rate, do you use to largest number compared to the value when x=0. For example the problem I am working on is: Your grandfather ...
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2answers
185 views

Prove: matrix A is diagonalizable iff exp(A) is diagonalizble

I need to prove: matrix A is diagonalizable iff $\exp(A)$ is diagonalizble. exp means exponent function. I know to prove that if $A$ is diagonalizable so $\exp(A)$ is diagonalizable, but have a ...
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1answer
66 views

how to solve equation $x^x=5$ [duplicate]

How can I calculate the equation $x^x=5$ Is it an exponential function? Thank you.
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5answers
110 views

solve the equation using logarithms (I think this is easy level)

Solve the equation for $x$ by using base 10 logarithms. $$16\cdot4^{2.5x}=9$$ EDIT: I made a typo (somehow... I was very far off!!) The correct equation is this: $$16\cdot4^{2.5x}=70$$ Can it be ...
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1answer
29 views

Finding best fitted value for power function. please help!

I need to find: 1. the best fitted value for $a$ in the power function 2. the best fitted value for $b$ in the power function Data given: I know that $b=bi$ and $a=e^{bo}$ --> my question is how ...
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1answer
26 views

List results of exponentiation, with natural bases and exponents

I am looking for a way to construct an ordered set like $\{2^3, 2^4, 3^3, 2^5, 2^6, 3^4, 5^3, 2^7...\} = \{8, 16, 27, 32, 64, 81, 125, 128...\}$ Preferably, but not necessarily, with all bases ...
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1answer
32 views

Solving this equation

Question: Solve: $$3^{2x^2}-2\cdot3^{x^2+x+6}+3^{2(x+6)}=0$$ I thought that we can take $a=3^{x^2}$ and $b = 3^{x+6}$. Then equation becomes $a^2-2ab+b^2=0$, which obviously means $a-b=0$. ...
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2answers
35 views

Which of the following is the highest value?

Question: Find the highest value among $12^9$, $10^{11}$ and $11^{10}$. I have seen problems like this, but they had surds, these are integers. Also, the LCM of $10$, $11$, $9$ $(990)$ is fairly ...
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2answers
19 views

Logarithms involving decimals

I am a student wondering how would I put this correctly into a calculator. I have 1,05 and 1,216 1,05^n=1,216 How would I calculate n without just multiplying 1,05 against itself until I hit the ...
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2answers
41 views

How to integrate $a^{-x}$

How to integrate $a^{-x}$. This is from a text book: $\int\frac{1}{a^x}dx, \text{a is a constant} $ I really can't think of a way of doing this, but the book says it converges using integral test, ...
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1answer
108 views

Solve for x when $2222^{5555} + 5555^{2222} \equiv x \pmod{7}$ [duplicate]

I need to find the remainder when $2222^{5555} + 5555^{2222}$ is divided by $7$. I'm thinking that Fermat's Little Theorem might help. Any suggestions?
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2answers
51 views

mental math: approximating fractional exponents

Does anyone have any good tricks for estimating expressions with fractional exponents (besides guess and check)? For example, I want to easily calculate $9.1^{1/3}$. Currently, the best I've got is ...
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2answers
21 views

Exponential equation help

Solve $x^{ln x} = e^{(lnx)^{3}}$ I'm looking at the mark scheme but I don't understand what they've done. I'd appreciate it if someone could explain every step. MS: taking ln of both sides or ...
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0answers
32 views

Powers of (large) lower triangular matrix

Consider the following "game" of chance. Each time the player pushes a button he is awarded a random (finite, integer, non-negative) number of points. The probability of receiving any particular score ...
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4answers
79 views

How to find the integral $\int4^{-x}dx$?

What approach would be ideal in finding the integral $\int4^{-x}dx$?
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2answers
78 views

Generalized power rule for derivatives

Background This background is not really necessary to answer my question, but I included it here to provide context. This question has some programming aspects to it as well, but since my question ...
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1answer
17 views

Variable Base with Variable as Factor in Exponent, Find Value

I saw a problem recently that looked like this: Assume $w$ and $z$ are positive. If $z^{4w} = 64$, what does $z^{6w}$ equal? And I had absolutely no idea how to even begin attempting this equation. ...
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3answers
41 views

How to find exponent of a number in a combination?

How do I find the exponent of $7$ in $^{100}C_{50}$ that is, $\dfrac{100!}{(100-50)!\cdot 50!} =\dfrac{100!}{50!\cdot 50!}$, this question was out of the blue, and I haven't been able to find any ...
0
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2answers
47 views

rational exponents. two differing answers.

This is not homework. Example 3) (d) of section P.4, rational exponents in Algebra and Trigonometry: $$\frac{1}{\sqrt[3]{x^4}} = \frac{1}{x^\frac43} = x^{-4/3}$$ Completely rational. Almost ...
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1answer
43 views

Is there a convention for power of a half being the positive square root?

I know the $\surd$ sign refers to the positive square root. Does the exponent 1/2 mean the positive square root too by convention? I ask because I'm converting from parametric to cartesian here... ...
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1answer
42 views

Homework help to rearrange formula

Given the equation $${V_m} = u(\ln {m_0} - \ln {m_8}) - g{t_f}$$ I need to solve for ${m_0}$ Here is what I have but it looks messy and I feel like there is sometihng wrong or a better way 1st ...
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2answers
39 views

Reciprocal of $7.5^{1-x}$

Ok my calculator tells me the reciprocal of $7.5^{1-x}$ is $0.1333\cdot7.5^x$. Can anyone explain the steps involved to get this manually? Is it along the line of the reciprocal of $7.5^1 + 7.5^{-x} ...
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2answers
30 views

$3^a\mid s(n) \Rightarrow 3^a\mid n$

This is not a homework question, neither a championship problem (as far as I've searched in the net), and it came up noticing a singular pattern, involving the powers of $3$: "Prove or disprove that ...
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1answer
25 views

How does $10^{100}$ = $2^{\frac{100}{\log2}}$?

Googol is equal to $10^{100}$. To determine the number of bits that it needs to represented in binary, we need to rewrite Googol with a base of $2$. This is the correct answer: $$10^{100} = ...
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0answers
41 views

Justification for exponents other than positive integers

Here's a question that's bothered me ever since highschool, and I've never heard a good answer. I know that mathematicians can define operators to mean whatever they want, as long as their system of ...
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2answers
39 views

where does the modulus go when cancelling $e$ and $\ln$ in this problem?

So I did this problem today: Show that $\frac{dy}{dx} = yx^2$ can be written as $y = Ae^{\frac{x^3}{3}}$ my solution is shown below: $$ \frac{dy}{dx} = yx^2 $$ $$ \frac{1}{y} dy = x^2 dx $$ $$ ...
3
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1answer
71 views

On the equation $\exp(a x+b)=\ln(x)$

I am confronted with: $$\exp(a x+b)=\ln(x)$$ for $a,b$ reals and $a<0$, $b>0$. I need the (unique) solution for $x$. My first target is (if it exists) an analytic solution in terms of ...
0
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2answers
22 views

Find the largest exponent

I've got this GRE math question: The integer y is positive. If $6^y$ is a factor of $(2^{14})(3^{24})$, then what is the greatest possible value of y? The answer is ...
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0answers
21 views

Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $Dq(x) . Ax < 0$ for all $x \neq 0$

Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $$Dq(x) . Ax < 0$$ for all $x \neq 0$ Definition: a linear system $x' = Ax$ called ...
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1answer
64 views

Conditions required for $(z_{1}z_{2})^{\omega}=z_{1}^{\omega}z_{2}^{\omega}$, where $z_{1},z_{2},\omega\in\mathbb{C}$

I am having trouble finding the conditions on $z_{1}$ and $z_{2}$ in order for: $$(z_{1}z_{2})^{\omega}\equiv z_{1}^{\omega}z_{2}^{\omega}\qquad \forall\omega\in\mathbb{C}$$ My first step was to ...
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5answers
290 views

Why does $n^0 = 1$?

Why is it that $n^0 = 1$? I understand how $n^2 = n*n$ and how $n^1 = n$ but I can't understand why $n^0 = 1$.
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4answers
36 views

Find $m$ and $n$

Two finite sets have m and n elements. Thew total number of subsets of the first set is 56 more than the two total number of subsets of the second set. Find the value of $m$ and $n$. The equation ...
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2answers
50 views

Simple question on exponentiation

I know this one is trivial, but I was wondering: if I have something like $$a^{b^c}$$ then i know that it should be read as $$a^{\left(b^c\right)}$$ if no other parenthesis is present. Question: if ...
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2answers
22 views

Difference between growth formulas

What is the difference between $$N = N_0 \cdot e^{kt}$$ and $$N= N_0(1+r)^n$$ I'm trying to find the best formula to calculate population growth and sources seem to vary between these two?
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4answers
27 views

solve for x, giving answer to 3s.f?

I need help solving the question below: $$ 2x^ \frac{1}{4} = \frac {64} {x} $$ I know the answer is 16 but I'm not sure how to get to it. Can you explain how to get the answer so I can solve similar ...
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2answers
32 views

What is the 'growth constant'?

I'm looking into the formula of growth, namely $$N= N_0 e^{kt}$$ where $k$ is the 'growth constant'. What is the growth constant and how do I find it? I'm looking at a bug that has on average 1,67 ...
2
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2answers
64 views

compute the integral using residue theory

I am trying to compute an integral in an example in my complex analysis textbook: $$\int_{-\infty}^\infty {xsinx\over x^4+1}dx$$ The book gives some startup hints, but I don't quite follow, I set ...
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2answers
56 views

Is there a simple algorithm for exponentiating large numbers to large powers?

I've been thinking about this for some days, a multiplication is a lot of sums, so: $$75\times 75=\overbrace{75+75+75+75+75+75+75+75+\cdots}^{\text{75 times}}$$ But then, there is a simple algorithm ...
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3answers
48 views

Error raising a complex number to a power

I am trying to do $(3+7i)^5$ which acording to WolframAlpha and Mathway should be: $23028−11228i$ Yet I instead get: $6123+14287i$ -- I'm getting that answer by doing: $3^5 ...
2
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2answers
42 views

magnitude of complex exponental always equals 1?

as we all know $$e^{j\theta} = \cos\theta + j\sin\theta \\ |e^{j\theta}| = \sqrt{\cos^2\theta + \sin^2\theta} = 1$$ That means $|e^{j\theta}| = 1$ with any value $\theta$ is ($2\pi, \frac{\pi}{3}$, ...
2
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3answers
57 views

Finding the matrix exponential

Find the matrix exponential of $$\begin{bmatrix}1& 1\\ 0& 1\end{bmatrix}.$$ Since this matrix is not diagonalizable, you will have to use the definition of the matrix exponential. ...