Questions about exponentiation

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2answers
48 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 ...
2
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2answers
124 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 ...
1
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2answers
39 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 ...
5
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2answers
203 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 ...
-1
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1answer
70 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
115 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 ...
0
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1answer
32 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 ...
1
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1answer
28 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 ...
1
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1answer
36 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 $a-b=0$. ...
3
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2answers
36 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
24 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, ...
2
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1answer
138 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?
2
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2answers
83 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 ...
0
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2answers
23 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 ...
2
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0answers
45 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
81 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
173 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 ...
0
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1answer
20 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
45 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
49 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
47 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... ...
0
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1answer
62 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
29 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
56 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
42 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
24 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 ...
1
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1answer
66 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 ...
8
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5answers
294 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
42 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 ...
0
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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 ...
1
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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?
0
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4answers
28 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 ...
0
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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
73 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
64 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 ...
0
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3answers
50 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
49 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
62 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. ...
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2answers
31 views

Power calculation for simplification?

I have this simple question I saw here: ±(2 - 2^(-23)) × 2^128 = ±6.8 × 10^38 How did they get to ...
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3answers
42 views

question on surds i already asked this question but the answer I got did not match the one in the book [duplicate]

$$\sqrt{ 3x }= x + \sqrt {3}$$ Give x in the form $$A \sqrt {B} + C $$ Can you show me how this is done step by step. The answer I have in the book is: $$\frac {1}{2} \sqrt{3} + \frac {3}{2} $$ ...
0
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2answers
53 views

If $\sqrt{x+y}+\sqrt{y+z}=\sqrt{x+z}$, then $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=?$

If $\sqrt{x+y}+\sqrt{y+z}=\sqrt{x+z}$, then $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=?$ I really am stumped on this problem. I squared the first equation and found that $-y = \sqrt{(x+y)(y+z)}$. So ...
4
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3answers
758 views

how to calculate 2^1.4

So I have got a very basic question but it didn't come up as a google search so I am posting it here. I want to know how to easy calculate 2^1.4 = 2.6390... ...
0
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2answers
38 views

Subtracting 2 fractions with variables in the denominator that have different exponents.

Sorry for the relatively elementary question, but I am having trouble remembering exactly how to do this type of problem. I am looking to simplify this: $$ \frac{3}{4t^{1/4}} - \frac{1}{2t^{3/4}} $$ ...
1
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2answers
41 views

Inequality with trigonometric functions

Find all values for $a$ such that the following inequality holds: $$\sin^6x + \cos^6x + a\sin x \cos x \ge 0$$ To be fair, I didn't manage to get anything helpful wiht my calculations. I tried to ...
0
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0answers
34 views

What is the general notation for the principal value of complex exponential?

It is general to distinguish the principal value of complex logarithm set by denoting it $Ln( z)$. Is there any general notation to distinguish the principal value of complex exponential? In complex ...