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

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0
votes
2answers
64 views

Find function which satisfies the initial value problem: $\displaystyle 6 \sec x \,\frac{dy}{dx} = e^{y + \sin x} $

Find the function which satisfies the initial value problem: $\displaystyle 6 \sec x \,\frac{dy}{dx} = e^{y + \sin x} $ $\displaystyle \;\ y(0) = -9 $ So as far as I understand it, I should move ...
6
votes
3answers
90 views

Showing that $x^x/(2x)!=0$ as $x$ approaches infinity

How would I show $$\lim_{x\to\infty} \frac{x^x}{(2x)!}=0$$ I know $x^x$ grows faster than $(2x)!$ So then would I do $$\frac{x^x}{2x(2x-1)(2x-2)(2x-3)\cdots(2x-(2x-1))}$$ But how do I proceed.
1
vote
2answers
67 views

The value of $0^n$

In an attempt to compute $\sum_{k=0}^{n}{(-1)^k {n\choose k}}$ the author put $x=-1$ in the formula $(x+1)^n=\sum_{k=0}^{n}{ {n\choose k}x^k }$ Then he wrote that we have $$0^n = \begin{cases} 0 ...
0
votes
3answers
57 views

How to simplify $2^{\log_2 M}$? [closed]

How to simplify $2^{\log_2 M}$? I am aware the solution should be $M$, but unsure how to derive that.
1
vote
1answer
23 views

Predictability of consecutive powers

If $$a^b=c^d+1$$ $$a,b,c,d\in Z$$ $$a,b,c,d>0$$ then $a^b$ and $c^d$ are defined to be "consecutive powers" (atleast in my Q). Given two large positive integers $a$ and $b$, is it ...
2
votes
1answer
23 views

A modular arithmetic exponent problem

I was learning modular arithmetic and there was a problem that $4^{62} \equiv \ ? \pmod{7}$ The solution was to find the mods of the same number's exponents starting from one until the result equals ...
2
votes
1answer
30 views

Level curves of powers of a sphere

If we have $f(r) = r^n$, where $r^n = (x^2+y^2+z^2)^{n/2}$, and one wishes to examine the level curves of $f(r)$, namely $f(r) = k$ for some real number $k$, then one might proceed as follows: there ...
-1
votes
1answer
115 views

Exponentiation and far too high numbers?

I love very, very, very, big numbers! You see, I'm working on powers of $2$ and I need to calculate the next expression in this sequence: $2^2=4$ $2\uparrow\uparrow2=216$ ...
2
votes
3answers
109 views

An interview question: $2.1^{3.1}$ vs $3.1^{2.1}$,$ 2.1^{4.1}$ vs $4.1^{2.1}$, which is larger?

While Mathematica told me that $2.1^{3.1} - 3.1^{2.1} = -0.786932$ and $2.1^{4.1} - 4.1^{2.1} = 1.58855$, I wonder how to compare them quickly, by hand. I see $2^3 < 3^2$, so perhaps we have ...
5
votes
1answer
43 views

Not a root of unity - how to prove?

How does one prove that a given complex number is not a root of unity for some positive integer power n? Say, I want to prove that there does not exist a positive integer $n$ such that $(2i)^n = 1$, I ...
-1
votes
1answer
39 views

Last question about logs.

QUESTION Solve: $$\log_{\frac 1 2} (x+1)^{ \log_2x+1}=0.3$$ Attempt $\frac {\log( x+1)}{\log 2}$ multiply $\frac{\log (x+1)}{\log 2^{-1}}$. Sorry! but thanks.
0
votes
3answers
27 views

Quick Log problem?

$2^{(x^3)} = 3^{(x^2)}$ Solve for x I'm pretty sure I use logs to solve this, but how? to what base? I'm kinda lost.. Thanks
4
votes
5answers
143 views

Solve for $x^{11} + \frac{1}{x^{11}}$ if $x^3 + \frac{1}{x^3} = 18.$

Solve for $x^{11} + \frac{1}{x^{11}}$ if $x^3 + \frac{1}{x^3} = 18.$ I'm familiar with variants of this problem, especially those where the exponent in the given is a factor of the exponent in what ...
0
votes
1answer
48 views

Calculate power of irrational number modulo of integer

There are very efficient ways to calculate powers of integers modulo an integer, one of them is implemented by the Python pow function. I need to calculate ...
1
vote
1answer
140 views

Is it known whether ${\sqrt{2}}^{\sqrt{2}}$ is irrational? [duplicate]

I know the famous proof that uses $x={\sqrt{2}}^{\sqrt{2}}$ to prove that there must exist an irrational to an irrational power that evaluates to a rational. But I don't know if $x$ itself is known to ...
2
votes
5answers
78 views

Calculating very large powers of $e$

I need to calculate $e^{1763.192674118048}$, but when I try to calculate it directly using Matlab it returns "Inf", i.e. it can't calculate it. How do I calculate this? For what it's worth, just ...
6
votes
2answers
190 views

How to properly apply the Lie Series

I am trying to solve this problem from Symmetry Methods for Differential Equations A Beginner's Guide (Peter E. Hydon). Use the Lie Series ...
2
votes
1answer
59 views

Solving $(5-2\sqrt6)^{x/2}+ (5+2\sqrt6)^{x/2} = 10$

I know the answer is $2$. I guessed it. But how do you do it mathematically? $$(5-2\sqrt6)^{x/2}+ (5+2\sqrt6)^{x/2} = 10$$
6
votes
3answers
235 views

How to solve $B = x^c - (1 - x)^c$

How to solve for x ? Where we are interested in the range $0 < x < 1$ and $C \neq 0$. $$ B = x^c - (1 - x)^c.$$ The only thing I could come up with is to substitute $$ x = \sin^2y ...
2
votes
2answers
68 views

What is the difference between $-1^2$ and $(-1)^2$? [duplicate]

Intuitively, I though that $-1^2$ and $(-1)^2$ were exactly the same thing; however, it seems I was wrong as Wolfram Alpha (and any other calculator) returns $-1$ for the first case and $+1$ for the ...
-1
votes
1answer
71 views

How to solve for $n: n \le 2^n a$?

How to solve this equation for $n$: $n \le 2^n a$? Note: $a$ is a positive constant. My Method: $\\ n \leqslant 2^n a\\ \lg {n} \leqslant \lg{(2^n a)}\\ \lg{n} \leqslant \lg{a} + \lg{2^{n}}\\ ...
2
votes
1answer
84 views

Positive integer solutions to $a^x-b^y=2$

Does anybody know the all positive integer solution of $$ a^x-b^y=2 $$ under the condition $$ x \geq 2, y\geq2 $$ I didn't find any solution without $$ x=5,y=3,a=2,b=3 $$
3
votes
3answers
132 views

Show that $(2^n-1)^{1/n}$ is irrational

How to show that $(2^n-1)^{1/n}$ is irrational for all integer $n\ge 2$? If $(2^n-1)^{1/n}=q\in\Bbb Q$ then $q^n=2^n-1$ which doesn't seem right, but I don't get how to prove it.
0
votes
2answers
23 views

Simplifying ${2\sqrt[3]{e} \times 4\sqrt[3]{e^2}}$

Simplify this expression: ${2\sqrt[3]{e} \times 4\sqrt[3]{e^2}}$ The answer is apparently ${8e}$ I can see that ${\sqrt e^2}$ is ${e}$ but I don't understand the reasoning of why the cube root ...
0
votes
1answer
39 views

What is the difference in exponential of log and ln

What is the difference in exponential of log and ln? For example, exp(ln(sqrt(2)) and exp(log(sqrt(2)) What will be the answer ...
-1
votes
3answers
61 views

Is it valid to simplify $\sqrt{(t-7)^3}$ to $(t-7)^2$?

$\lim\limits_{t \to 7^+} \dfrac{\sqrt{(t-7)^3}}{t - 7}$ Can you simplify the top of the quotient to   $(t-7)^2$ ? Or a more general question, does the square root of exponent 3 simplify to ...
1
vote
2answers
97 views

How to evaluate $45^\frac {1-a-b}{2-2a}$ where $90^a=2$ and $90^b=5$ without using logarithm?

Let $90^a=2$ and $90^b=5$, Evaluate $45^\frac {1-a-b}{2-2a}$ I know that the answer is 3 when I used logarithm, but I need to show to a student how to evaluate this without involving logarithm. ...
0
votes
1answer
28 views

Prove the following using induction on d (matrices)

I manage to reach the step where I need to prove n = k + 1 but I am battling to complete the proof as I am not certain what to do with the exponents in my answer. I will run through the proof as I ...
0
votes
3answers
39 views

Relation between $x,y,z$…Exponent problem…

The given equation is- $\sqrt[x]{75} = \sqrt[y]{45} =\sqrt[z]{15}$ Now,it is required to prove $x+y=3z$. I want the simplest possible solution.Thanks in advance.
2
votes
4answers
138 views

If $3^{33}+3^{33}+3^{33}=3^{x}$. Solve for $x$.

If $3^{33}+3^{33}+3^{33}=3^{x}$. Solve for $x$. So we have: $$3^{33}+3^{33}+3^{33}=3^{x}$$ I added the left side and obtained: $3(3^{33})=3^{x}$ The problem I have is that extra $3$. If not, I ...
14
votes
4answers
3k views

How to find out which number is larger without a calculator?

So I have a question which is: Which is larger? $$2.2^{3.3} \text{ or } 3.3^{2.2} $$ Now I need to find out with using a calculator but the answer is $3.3^{2.2}$. The only thing I could think of ...
4
votes
4answers
103 views

Matrix exponential: $\begin{pmatrix} 0 & 1 \\ -4 & 0 \end{pmatrix}$

It is asked to calculate $e^A$, where $$A=\begin{pmatrix} 0 & 1 \\ -4 & 0 \end{pmatrix}$$ I begin evaluating some powers of A: $A^0= \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\; ; ...
5
votes
1answer
53 views

Let $a$ and $m$ be positive integers such that gcd$(a,m)=1$. Show that: $a^m+1$ is not a prime.

Let $a$ and $m$ be positive integers such that gcd$(a,m)=1$. Show that: $a^m+1$ is not a prime. Though I didn't check the statement with so many integers, but it looks like the equation never ...
7
votes
3answers
856 views

Exponent of an exponent?

If I have an expression that gives 2^3^4, would I compute this as $(2^3)^4$ or as $2^{(3^4)}$? The two answers are wildly different. My TI gives the former but Wolfram gives the latter and I don't ...
0
votes
0answers
30 views

Finding n for a given P of a Bernoulli trial

I'm randomly sampling $N$ items and I want to find $n$ such that I have a probability $P$ that I'll miss one. Practically, I'd select $P$ to be something like $10^{-12}$ so I'm almost assured to ...
1
vote
0answers
34 views

Simplify $e^{x \cdot \log{y}}$ where $x, y \in R^N$

I'm looking to simplify the following expression (or to determine if it's even possible). Given two vectors $x, y \in R^N$, simplify $e^{x \cdot \log{y}}$. I found it in some m-code for an infinite ...
5
votes
2answers
72 views

Does every prime of the form $4k+1$ divide a number of the form $4^n+1$?

While playing around with Fermat's little theorem I was asking myself the question in the title and I can't answer it...
0
votes
5answers
74 views

Are these two expression equal?

My friend insisted that $(-1)^{(-n)}$ is equivalent to $(-1)^n$ for any number of $n$. A quick check in the Wolfram Alpha show ...
16
votes
3answers
2k views

How to solve equations to the fourth power?

Is it possible to manually retrieve the value of $y$ from the following equation $$\color{blue}{153y^2-y^4=1296}$$ WolframAlpha has four solutions for $y$: $-12, -3, 3, 12$. How has it solved? What ...
35
votes
2answers
1k views

Proving that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator

Prove that $e^{\pi}-{\pi}^e\lt 1$ without using a calculator. I did in the following way. Are there other ways? Proof : Let $f(x)=e\pi\frac{\ln x}{x}$. Then, ...
1
vote
3answers
175 views

Will $a^a$ ever out-grow $9^{9^{^\ldots}}$?

I am trying to come up with the largest finite number that can be made using a set number of characters. I have two expressions which are calculated and printed out by a program (theoretically - they ...
3
votes
1answer
86 views

A symmetric system of nonlinear equations - how to solve?

So, I was adviced to ask a new question on my problem (as the first one wasn't very precise), that is to solve the system of equations: $$\begin{cases} x\cdot y=6 \\ x^y+y^x=17 \end{cases}$$ where: ...
1
vote
1answer
77 views

If $x^{x^4}=4$. Find $x^{x^2}+x^{x^8}$

If $x^{x^4}=4$. Find $x^{x^2}+x^{x^8}$. I found this one in a competitive exam paper and found it interesting. Thanks for any help.
6
votes
1answer
150 views

Does $(1^a+2^a+3^a+4^a+5^a)^b=1^c+2^c+3^c+4^c+5^c$ imply $(a,b,c)=(1,2,3)$?

Question : Is the following proposition true? Proposition : For positive integers $a,b,c$ where $b\ge 2$, if $$(1^a+2^a+3^a+4^a+5^a)^b=1^c+2^c+3^c+4^c+5^c$$then $(a,b,c)=(1,2,3)$. This is ...
1
vote
2answers
62 views

Simplify $\,\sqrt[10]{32a^5}$

I'm not sure if this is the correct site to ask such an elementary question but I'm trying to teach myself basic algebra and I can't understand how to do this one equation it's been so annoying. So ...
22
votes
13answers
938 views

Which is greater, $98^{99} $ or $ 99^{98}$? [duplicate]

Which is greater, $98^{99} $ or $ 99^{98}$? What is the easiest method to do this which can be explained to someone in junior school i.e. without using log tables. I don't think there is an ...
-1
votes
4answers
84 views

Calculate the power, given all other numbers in an equation

$$100 = 200(2)^x$$ Given all numbers in the equation, how do I find $x$?
0
votes
3answers
173 views

Limit of a function raised to a power

I was working with extraction of non-electrolytic solutions and was sketching a mathematical formulae to find the limit of extracting a solvent by Nernst equation when I stumbled on this limit. ...
1
vote
4answers
133 views

Last two digits of $3^{7^{2016}}$

I need help with solving this Algebra problem: Find the last two digits of $3^{7^{2016}}$. Preferably using Euler's theorem.
2
votes
1answer
206 views

X raised to power-X raised to power-3 equals to 3.

The question is what are the possible values of $x$ when we have $$x^{x^3} = 3$$ (that is $x^3$ in the exponent itself and not $x*3$). I solved one answer by guessing that $x = \sqrt[3]3$. My work ...