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

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1answer
73 views

Formula for the number of digits in the number $2^x$

I'm wondering if there is a formula for the number of digits in $2^x$. For example if $x = 3$ then the number of digits is equal to $1$ because $2^3 = 8$ or for example if $x = 4$ then the number of ...
2
votes
1answer
412 views

The domain of fractional exponents

Take the following: $$f(x) = x^{6/4}$$ The domain of this function is all real numbers. This function can be simplified to: $$f(x) = x^{3/2}$$ The domain of this function is all real numbers ...
1
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2answers
122 views

Checking inequality without actually calculating LHS and RHS

How to check whether the following inequality is true or not without actually calculating the values of $x^y $ and $y^x $: $$ x^y > y^x$$ (x and y are integers)
3
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3answers
142 views

Can a fourth-order equation be solved like a quadratic equation?

I was asked to find the zeros of $y = x^4 + 5x^2 +6$. I tried to turn this into a quadratic to factor it as follows: $y = x^4 + 5x^2 +6 = {(x^2)}^2 + 5{(x^2)}^1 + 6$ Put another way: Let $t = ...
0
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2answers
159 views

Rational exponents: prove some states

In some rational exponent expressions the solution isn't a real number why? Example (explain what I mean): $$\begin{align} \Big(-x\Big)^{1/n}=\left\{\text{is not a real number}\right\} \end{align}$$ ...
1
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1answer
48 views

Is the statement $a+b < c+ d \implies e^{-a} + e^{-b} > e^{-c} + e^{-d}$ true?

As the title says, I am trying to ascertain whether the following is true: Suppose $a,b,c,d\in \mathbb{R}^+$ are such that $a + b < c + d$, then it is also true that $e^{-a} + e^{-b} > e^{-c} + ...
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2answers
73 views

Why is $0^0=1$, given the following information? [duplicate]

Why is $0^0=1$, given the following information? We really have two separate rules that are at odds with each other. Typically we have $0^n=0$ (provided n is positive) and $a^0=1$. Each of these ...
0
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2answers
91 views

How to find the remainder of $(2010^{1020} + 1020^{2010})$ divided by $3$

What is the remainder when $2010^{1020} + 1020^{2010}$ is divided by 3?
2
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4answers
230 views

How to find $\sqrt{1+{4\over x}+{4\over x^2} }$?

If $$abx^2 = (a-b)^2(x+1)$$ then what is $$\sqrt{1+{4\over x}+{4\over x^2} }$$ (A) $a+b \over a-b$ (B)$a-b\over a+b$ (C) $a^2+ab$ (d) None EDIT: What I've done is this: ...
1
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1answer
84 views

How to solve this equation with parameters on power?

Please let me know how to solve this equation: $$100^{1-b}=\frac{1}{2}40^{1-b}+\frac{1}{2}200^{1-b}$$ I try to use the trick of $x=e^{\log x}$ But it doesn't work And $b\not=1$
8
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1answer
166 views

Need a general formula for $\frac{d^n}{dx^n}\left(f(x)^m\right)$

Let $m,n\in\mathbb{N}$. I need to express the derivative $\displaystyle\frac{d^n}{dx^n}\left(f(x)^m\right)$ in terms of sums/products of the derivatives of the function $f$ itself. Here are results ...
1
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2answers
36 views

threshold of n to satisfy $a^n <n^a$

How to find the minimum of $n$ when we know $a$, to satisfy: $a^n<n^a$ $a^m>m^a$ for each $m>n$ $n$ and $m$ are natural numbers.
0
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1answer
43 views

Calculate the Burning Time for a Lamp

If you have a lamp with burning time 4000 hours. If the time goes forward until the lamp will be destroyed the exponential distribution is 3675 hours, what is the probability of a lamp to be working ...
2
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1answer
64 views

How to find $x + y + z$?

Q. If $x^{1/3} + y^{1/3} + z^{1/3} = 0$, then (A) $x + y + z = 3 xyz$ (B) $x + y + z = 0 $ (C) $( x + y + z)^3= 27 xyz$ (D)$ x^3 + y^3 + z^3 = 0$ What I've done: ...
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0answers
1k views

How to solve polynomial-exponential equation

I'm trying to solve equations like the following one: $$5 + 3x - 4x^3 = e^{x^2}$$ I've tried using the Lambert W function, but I didn't get any success. I must admit I'm relatively new to Lambert W ...
5
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1answer
72 views

Can this type of limit even be evaluated?

This is going to be a long question; please bear with me. We are familiar with the notations $\Sigma^{n}_{k=0} a_k$ and $\Pi^{n}_{k=0}a_k$ for the sum and product of the finite sequence $\{a_n\}$. ...
1
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2answers
205 views

Exponential equation with absolute value: $9^{|3x-1|}=3^{8x-2}$

$$9^{|3x-1|}=3^{8x-2}$$ Can someone show me the steps on how to solve this, i've been trying for 30 minutes
5
votes
2answers
201 views

Fraction raised to integer power

if I have $(p/q)^n$ where $p,q,n$ are integers and $p/q$ is a... I don't know what you call it. Not a whole number, but something like 15/7 where you can't reduce it any more and it's non-integer. Can ...
2
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0answers
214 views

Solving $x$ for $y = x^x$ using a normal scientific calculator (no native Lambert W function)?

Solving $x$ for $y = x^x$ using Lambert W function is clear enough thanks to this handy answer, but as I'm using the solution in a network support document I need it in a form that can be solved on a ...
0
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1answer
62 views

Relation of $e$ to other numbers…

I found the following result, When i was working on my calculator . $$x^y < y^x \quad ,x < y \quad \text{ for } x,y<e$$ $$x^y > y^x \quad ,x < y \quad \text{ for } x,y>e$$ I can't ...
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1answer
62 views

How do I simplify this expression?

How do I simplify this expression? $$\frac{(4-x)^2 (1/3) (6x+1)^{-2/3} (6) - (6x+1)^{1/3} (-2x)}{(4-x^2)^2}$$
0
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1answer
187 views

How do I simplify the expression (a^-1 + b^-1) ^-1?

How do I simplify the expression ... $$(a^{-1} + b^{-1}) ^{-1}$$ ?
0
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1answer
42 views

Simplifying exponentials of the form $\,a^x \cdot b^y$

I am given the exponential $\left(\dfrac{1}{2}\right)^x\cdot 4^{(x/2)}$. While my intuition screams that this can be simplified to $\dfrac{2^x}{2^x} = 1$, I am unable to see a concrete mathematical ...
1
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0answers
48 views

Matrix exponent and representations of $\mathbb{R}$

It is well known that $\exp(C^{-1}AC)=C^{-1}\exp(A)C$, where $C$ is an invertible matrix. Really, $$\exp(A)=\sum{}\frac{A^k}{k!} \quad \text{and} \quad (C^{-1}AC)^k=C^{-1}A^kC,$$ so the first ...
0
votes
2answers
41 views

Simplify the expression.

Simplify the following expressions using fractional exponents, I forgot how to do this type, do I rationalize it? or can it just cancel each other out? $ {\Large \frac{\sqrt[ 5 ]{x^{ 3 }}}{ \sqrt{x} ...
0
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1answer
160 views

Simplify the following expressions using fractional exponents

Simplify the following expressions using fractional exponents. Display your answer using fractional exponents. $${ \sqrt[ 3 ]{x^{ 7 }} = }\text{ and }{ \sqrt[ 7 ]{x^{ 3 }} = }$$ Thanks..not sure ...
0
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1answer
60 views

What's the pattern? roots and powers of n?

I won't hold it against anyone if this is considered a bad question but I just don't know how else to put it and really want to know. 1st case: n = positive integer a = $\sqrt n$ b = 1 2nd case: n = ...
2
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0answers
92 views

Is $x^{\frac 12}$ the same as sqrt(x)

maybe this question is very simple and clear and trivial to everbody. but right now i'm not sure. the equotation $$ x^\frac{1}{2} = \sqrt{x} $$ is only true whenever $ x \geq 0 $ right? the square ...
8
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1answer
174 views

Expressing the values of a matrix at pow N

I have a square matrix (that comes from a Markov Chain) that looks like that: $$Q = \begin{bmatrix} 0 & 1& 0 & 0 & .. & 0 & 0\\ 0 & a & 1-a & 0 & .. & 0 ...
1
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1answer
84 views

Systems of equations question

\begin{align*}a^a\cdot b^b\cdot c^c\cdot d^d&=\frac12\\a+b+c+d&=1\end{align*} How can we find solutions for this system of equations given that $a, b, c, d > 0$ ?
2
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2answers
216 views

A non-exponentially bounded analytic function?

A function $f:\mathbb R\to\mathbb R$ is said to be exponentially bounded if there is an $n$ such that for sufficiently large $x\in\mathbb R$, $\exp(\exp(\cdots \exp(x)))>f(x)$ (where the $\exp$ is ...
10
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1answer
854 views

What is wrong with this funny proof that 2 = 4 using infinite exponentiation?

Out of boredom, I decided to recall the following equation: $$x^{x^{x\cdots}} = 2.$$ Which, I simply rewrote like this: $x^2 = 2$, and therefore $x = \sqrt{2}$. Then I took a look at the more ...
2
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2answers
47 views

$P[ e^{tX} > e^{ta} ] =?$

Can anyone help me understand why given any random variable $X$, the following stands true? $$ \forall t > 0, P( e^{tX} > e^{t\epsilon} ) \le e^{-t\epsilon} E[e^{tX}]. $$ I found it in the ...
1
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4answers
184 views

Exponent rules with negative numbers

Hello if you could answer this please what is the difference between -2^2 and (-2)^2
1
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1answer
99 views

How to solve strange exponential equation?

How would equations of the form $b^x-x^a=0$ be solved for $x$, given $a$ and $b$? For instance, specifically, how would $2^x=x^2$ be solved? Does a method exist?
1
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1answer
39 views

How is this possible? Can someone explain?

My teacher says that $W^{2/7}B^{5/7}=1$ is equivalent to $W^2B^5=1$. Can someone explain this rule to me? Am I always able to just take the variable and raise it to the numerator of the fractional ...
0
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1answer
21 views

Transformation. Take n out of a root

Im kinda confused. If n is always > 0 $$ (n-a)^{\frac{x}{y}} = n*(1-\frac{a}{n})^{\frac{x}{y}} $$ is that true? Because there were some transformations in recent answers to my threads where I did ...
0
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1answer
34 views

Growth rate of $n^2$ vs $(\log_3(n))^3$

Which grows faster, $n^2$ or $(\log_3(n))^3$? How do I figure out which grows faster in general in these kinds of situations?
1
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1answer
220 views

What comes after exponents?

We use multiplication for repeated addition, and in turn use exponents for repeated multiplication. What topic comes after this, for repeated exponentials? Is there something my teachers are hiding ...
5
votes
5answers
271 views

No difference between $0/0$ and $0^0$?

I have seen discussions about both $0/0$ and $0^0$ and they differ a bit in the way that most seem ok with calling $0/0$ "undefined", while the $0^0$ discussion still seems like a dispute. If this is ...
1
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1answer
37 views

Question on irreducible polynomials and primes.

Consider the polynomial $p(x) = 1+\sum_{i=1}^d a_i x^i$ where $a_i$ is binary and not all $a_i$ are $0$. Is it possible that $p(2^n)$ is prime for all integer $n>-1 ?$
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4answers
67 views

Exponential algebra problem

We need to solve for x: $$54\cdot 2^{2x}=72^x\cdot\sqrt{0.5}$$ My proposed solution is below.
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3answers
210 views

How to simplify $42^{25\sqrt{25^{25}}}$?

Am a student preparing for GRE, I have no clue to solve this am attaching the screenshot of question: I need you give me a short cut or tip to deal such problems...
17
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5answers
882 views

Do equal bases imply equal powers?

$$x^a = x^b \Rightarrow a =b$$ So, this is a concept I used in multiple math problems and they often turn out right. The thing is, today my math teacher told me that this is not necessarily true. ...
1
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2answers
79 views

Power correct notation

Ok, I know this may sound dumb, but I am trying to understand which is the correct (most beauty) notation for the power function ${\rm pow}(f(x),n)$. This is the correct one: $[f(x)]^n$ From ...
1
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1answer
45 views

About a matrix, its powers, and a particular value…

Is it possible to find two matrices, $A$ an $B$ such that: $$A B^0 x = \begin{pmatrix} c_0 \\ ? \\ ? \\ \end{pmatrix}, $$ $$A B^1 x = \begin{pmatrix} c_1 \\ ...
1
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1answer
89 views

Question about powers

I've been trying to solve this problem, but I can't do it by any means other than brute force, I need help, please. The result is: 6,000.00001 $$\frac{1}{10^{-3}}+\frac{10^2}{2\cdot ...
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1answer
407 views

RSA and calculating huge exponents

I am writing an Extended Essay on RSA encryption and in the essay, I am going through a worked example of all of the stages involved (key generation, encrypting and decrypting). I am using very small ...
7
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1answer
91 views

Finding $x^n$ patterns

I noticed the other day while computing consecutive powers of $2$ that for $n \geq 1$, the numbers in the ones place of the values of $2^n$ repeat every 4 terms $(2, 4, 8, 6,\ldots)$. In the tens ...
1
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2answers
206 views

Multiplying and simplifying expressions

The expression is: $$\frac{24a^4b^2c^3}{25xy^2z^5} \cdot \frac{15x^3y^3z^3}{16a^2b^2c^2}$$ What I did was subtract the exponents of the numerator to the exponents of the denominator. I did a cross ...