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

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0
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2answers
64 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 ...
1
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1answer
79 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... ...
1
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1answer
82 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
41 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} ...
2
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2answers
34 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 ...
1
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1answer
34 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} = ...
1
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2answers
307 views

How to use Fermat's little theorem to find $50^{50}\pmod{13}$?

I don't understand how to use Fermat's little theorem to find remainders e.g if we are asked to find remainder of $50^{50}$ on division by $13$, what is a and what is $p$ in the formula? Also I ...
1
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0answers
102 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 ...
1
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2answers
58 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
80 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
45 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 ...
2
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0answers
24 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
77 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 ...
4
votes
8answers
263 views

Calculate the last digit of $3^{347}$

I think i know how to solve it but is that the best way? Is there a better way (using number theory). What i do is: knowing that 1st power last digit: 3 2nd power last digit: 9 3rd power last digit: ...
9
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5answers
314 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 $n^0 = 1$.
1
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4answers
47 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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3answers
69 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
28 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
34 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
33 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
102 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 ...
0
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2answers
88 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
59 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
votes
2answers
57 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
83 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. ...
0
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2answers
43 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 ...
1
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3answers
57 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
117 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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4answers
3k 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
47 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
53 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 ...
2
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1answer
86 views

Does a solution exist where $p,q$ are odd primes and $p^a - q^b = p^c - q^d$ where $a > c > 1$ and $b > d > 1$

From my thinking so far, there is no solution. Is this an open question or is the answer well known? Here's my reasoning about this issue: If a solution exists, then: $$p^c(p^{a-c} - 1) = ...
1
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1answer
118 views

What does it mean when a matrix is to the (-1/2) power?

I'm reading a machine learning paper that uses a form of matrix normalization called symmetric divisive; given a matrix A and a diagonal matrix D derived from A, we define $$N=D^{-1/2}AD^{-1/2}$$ I am ...
2
votes
2answers
127 views

First derivative of multiplied powers

Wolfram Alfa shows $\frac{d}{dx}e^{4y} = 4e^{4y}$ but I do not understand how to get to that answer I have $e^{4y} = (e^4)^y$ So by the chain rule is it not the case that \begin{align} ...
1
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2answers
92 views

Exponential of a 3x3 lower bidiagonal matrix

I have a 3x3 matrix with non-zero entries ONLY along the main diagonal and the diagonal above. There are exactly two non zero diagonals in the matrix like this \begin{pmatrix} a & 0 & 0 \\ d ...
1
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1answer
27 views

Supremum of (e^(i z t) - 1)/z

i'm new here, so i'm not sure if this is the right place to ask this question: I know that the following holds true: $$ \forall\, t \in \mathbb{R} \; \forall\,x\in\mathbb{R}\setminus\{0\} ...
1
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1answer
197 views

weighted average with exponential weighting

I want to create weighted average, where weights depend on value of number. If I want exponential weights is this regular? $average = \log_e(\frac{\sum_{i=1}^n e^{v_i}}{n})$ Isn't it just average of ...
0
votes
2answers
63 views

complex expression to the power of a complex expression

I have a math exam tomorrow, and i am not sure with my solution for a exercise. can you please tell me if i am right. Question is: $$(1+i)^{(1-i)}$$ My solution is: $$\sqrt{2} e^{(i ...
0
votes
1answer
49 views

Why expression under root has to be positive?

I have function defined like this : f(x,y) = $\sqrt[127,5]{\frac{x^²+y^²-4y}{4x-x^2-y^2}}$ I thouth that domain is $4x-x^2-y^2 \neq 0$ but when I looked on wolfram, the domain is everything under the ...
3
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1answer
296 views

matrix exponential limit

I'm having litlle trouble here to prove the following statement: "Let $A$ an $n\times n$ matrix (real or complex). Prove that $$\lim_{n \to \infty} \left(I + \frac{A}{n}\right)^{n} = e^{A}.$$ Now ...
0
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1answer
223 views

Why are exponents not associative?

I ran into something that seemed odd to me today: exponents are not associative. The following equation sums that up: $$ 10 * 2^{5x} \not\equiv 20^{5x} $$ Why is this the case? Is there some ...
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3answers
42 views

Proof of $\lim_{n \to \infty} a^{1/n} = 1$ and $\lim_{n \to \infty}b^{f(n)} = b^{\lim_{n \to \infty} f(n)}$ where $a>1$

As title says, I am not sure what would be the proof of $$\lim_{n \to \infty} a^{1/n} = 1$$ would be where $a>1$. Also, how do you prove that $$\lim_{n \to \infty}b^{f(n)} = b^{\lim_{n \to \infty} ...
1
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1answer
76 views

Cauchy's integral formula used on circle

If $\gamma$ is a piecewise, smooth, positively oriented simple closed curve in $D$, then Cauchy's formula states that $f(z)=1/2\pi i\int_\gamma {f(a)\over {a-z}}$. My textbook also stated that for ...
0
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1answer
42 views

Name of numbers in “to the power of” and factorial calculations

In $4*5=20$ , $4$ and $5$ are multiplicands and $20$ is the product. What are the names / labels of the numbers in the following expressions? $2^3=8$ $4!=24$
0
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2answers
56 views

How to solve an exponential function with multiple addends

Our math teacher gave us the following exponential equation to solve: $3^x+10=2*7^x$ ...and I was stumped. Eventually, the solution given was to graph both sides and find their intersection using a ...
9
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2answers
414 views

Exponential of a function times derivative

Exponential of a derivative $e^{a\partial}$ is simply a shift operator, i.e. \begin{equation} e^{a\partial}f(x)=f(a+x) \end{equation} This can be easily verified from a Taylor series \begin{equation} ...
24
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8answers
2k views

How does the exponent of a function effect the result?

The $x^{2/2}$ can be represented by these ways: $$\begin{align} x^{2\over2}=\sqrt{x^2} = |x|\\ \end{align} $$ And $$\begin{align} x^{2\over2}=x^{1} = x\\ \end{align} $$ Which one is correct? And what ...
0
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1answer
197 views

Why 1.66x10e-27 is 1.66e-26

I am confused with this new way of writing exponential. For example I saw a question where person has changed 1.66X10e-27 to 1.66e-26. I am confused how -27 is reduced to -26 thanks
4
votes
1answer
91 views

Is there any way to analyze an absurdly large exponent?

On a recent Giant Bombcast, someone wrote in and asked an absurd question (as is usual for this podcast). In short, the question was: Given a 1080p TV, how long would it take to view every ...
2
votes
1answer
188 views

Permuting digits in a power of $2$

Does there exist a natural number $N$ that is a power of $2$ whose digits (in the decimal representation) can be permuted to a different power of $2$? Thoughts: If such a number $N$ exists, then ...