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

learn more… | top users | synonyms (1)

0
votes
2answers
19 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
22 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 ...
0
votes
3answers
30 views

Valid way to simplify this limit?

$\lim\limits_{t \to 7^+} \dfrac{\sqrt{(t-7)^3}}{t - 7}$ 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
48 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
20 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
34 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.
1
vote
4answers
123 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
80 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
44 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
767 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
24 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 ...
6
votes
2answers
68 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
62 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 ...
15
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 $$153y^2-y^4=1296$$ WolframAlpha has four solutions for $y$: $-12, -3, 3, 12$. How has it solved? What I've achieved ...
30
votes
1answer
695 views
+50

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, ...
0
votes
1answer
128 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
71 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.
-1
votes
0answers
98 views

Finding the value of $(x+y)$ when $x^x+y^y=31$

As the title suggests...the question is to find the value of $(x+y)$ when $x^x+y^y=31$.Is it possible to solve this question without trial-and-error method when only this much information given.Using ...
6
votes
1answer
132 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
51 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
764 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
45 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
46 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. ...
2
votes
4answers
113 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
84 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 ...
1
vote
1answer
47 views

Is the following solvable for x?

I have the following equation and I was wondering if I can solve for x given that it appears both as an exponent and a base: $[\frac{1}{\sqrt {2\pi}.S}.e^{-\frac{(x-M)^2}{2S^2}}-0.5\frac{1}{\sqrt ...
0
votes
1answer
40 views

repeated exponents sign

I'm wondering if there is a exponent version of $\sum$ or $\prod$ or I've even seen a big k used for repeated division. Is there a similar symbol for exponentiation and are there any useful ...
2
votes
0answers
69 views

Compute sum of large powers [closed]

I have the following problem. There is an array that contains values that are to be powers of $-2$. I need to calculate the sum of these powers. For example, if the array is $\{3,4,5\}$ I need to ...
3
votes
0answers
32 views

Comparing Large Exponents with different bases.

How to compare large exponents with different bases? Is there any way to roughly approximate their values? For example, sort the elements of list below based on their magnitude. ...
0
votes
1answer
23 views

Simple way to generate a sequence where the 1st number is $x_1$, the tenth number is $x_{10}>x_1$ and the 20th number is above some order of magnitude

I'm a bit stumped at the moment. I'm trying to generate a sequence where the first number is 1.9, the tenth is 3, and the 20th is between $10^{5-6}$ . There should be a function $f(x_1) = 1.9$, ...
-4
votes
3answers
43 views

Difference of powers of two [closed]

Is there a simple way (involving minimal calculations) to calculate $2^{987}-2^{986}=?$ Answer: $2^{986}$
0
votes
1answer
43 views

$p^3 = 2009 + 47 * 2^q$ where p and q are primes

Solve the ecuations $p^3 = 2009 + 47 * 2^q$, where $p$ and $q$ are primes. Fermat's little theorem could help.
1
vote
1answer
21 views

What kind of operation/rule was applied here?

Maybe this is a typo in our assignment and solution, but I can't tell. The question: The solution: What happened here with the minus signs in the first factor and in the exponent? Edit: the ...
0
votes
1answer
21 views

Will the absolute logarithm always produce the correct real result if one exists?

I'm a computer scientist, so my math skills are a bit rudimentary. The application I'm writing is more or less about solving equations. I'm only interested in real number solutions, so imaginary ...
1
vote
2answers
46 views

Simplification of $n^{1/\sqrt{\log n}}$

I would like to simplify this function, how can we do it ?
4
votes
7answers
135 views

How do i convince students in high school for which this equation: $2^x=4x$ have only one solution in integers that is $x=4$?

I would like to convince my student in high school level using a simple mathematical way to solve this equation: $$2^x=4x$$ in $\mathbb{z}$ which have only one integer solution that is $x=4$ . ...
7
votes
4answers
187 views

Why can the integral $\int_{x=0}^{\infty} x\mathrm{e}^{-\alpha x^2}\mathrm dx$ not be evaluated by parts?

Can the integral $\int_{x=0}^{\infty} x\mathrm{e}^{-\alpha x^2}\mathrm dx$ be evaluated by parts to show that $\int_{x=0}^{\infty} x\mathrm{e}^{-\alpha x^2}\mathrm dx= \frac{1}{2\alpha}$ I know that ...
1
vote
2answers
62 views

series function

We know that there are some series that can be written in short, for example: $$ \sum_{n=0}^\infty x^n=\frac{1}{1-x},\qquad |x|<1 $$ Is there similar function for $$ \sum_{n=1}^N x^{1/n} $$ or $$ ...
0
votes
4answers
116 views

Why is $(-1)^x=e^{i\pi x}$

I was recently taught exponentials and I decided to play around with negative bases, which they told me were not allowed. The obvious place to start was negative one, and, as expected, the graphing ...
0
votes
1answer
68 views

How to show that we reach $1$ at an odd or even turn without brute force

Consider the following challenge between two players A and B. They are given the initial terms $a_0= 3^{2014}$ and $b_0= 15^{4028}$ of two sequences, and the scope is to reach $1$ before the other, ...
3
votes
2answers
59 views

Quick Exponent Clarification

$N = 5^{\displaystyle 5^{\displaystyle 5^{\displaystyle 5^{\displaystyle 5}}}}$ In the following equation is N equal to $5^{5^4}$ or $5^{(5^{(5^{(5^5)})})}$? One of them is huge compared to the ...
25
votes
10answers
3k views

What is the accepted syntax for a negative number with an exponent?

A friend is taking a college algebra class and they are teaching him that $$-3^2 = -9$$ Their explanation is: $$-3^2 = -(3^2) = -9.$$ It has been a long time for me but I thought that in the ...
8
votes
2answers
151 views

An interesting property of binomial coefficients that I couldn't prove

So when I was trying to prove the argument in this link I've come up with something. When you extract the left term from the right term, you get the term under them. What is interesting is that as ...
1
vote
4answers
127 views

What is the value of $\lim_{x\to 0}x^x$?

Evaluate $$\lim_{x\to 0}x^x$$ I tried by writing $x$ in terms of exponentials: $x=e^{\ln x}$ so $x^x=e^{x\ln x}$ $\lim_{x \to 0}x \ln x=\lim_{x \to 0}(\ln x +1) =-\infty$ Thus $\lim_{x\to ...
1
vote
2answers
24 views

How to estimate magnitude of expontent?

When given an exponent, such as 6^12, is there a simple way to approximate how large(magnitude) the result is, without performing the calculation? Is this method accurate for large exponents?
2
votes
2answers
264 views

0's Exponents are impossible? [duplicate]

I've had something that's been bugging me, and I tried research and asked my math teacher. None had sufficient answers. The concept of $0$ is that when $0$ goes to any exponent except for $0$, it ...
-4
votes
1answer
26 views

Need Help: Exponential Equations (Same bases) [closed]

$3^{n+2} + [3^{n+3} - 3^{n+1}] = ?$ How do we get the answer for this? Do I just remove the bases and proceed to find the value of $n$ or do I use logarithms?