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

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7
votes
4answers
205 views

Find the value of $x$ if $x^{x^4} =4$

Find the value of $x$ if $x^{x^4}=4$. Given options are $2^{1/2}$ $-2^{1/2} $ Both 1. & 2 None of the above From option verification, we get option 3. as correct one. But is there any real ...
0
votes
2answers
37 views

Is it possible to convert $y = a^x + b^x$ to the form of $y = a \cdot b^x$?

I don't think this is possible, but I wanted to ask people who know more than me. Is it possible to convert $y = a^x + b^x$ to the form of $y = a \cdot b^x$?
25
votes
7answers
3k views

How do I compute $a^b\,\bmod c$ by hand?

How do I efficiently compute $a^b\,\bmod c$: When $b$ is huge, for instance $5^{844325}\,\bmod 21$? When $b$ is less than $c$ but it would still be a lot of work to multiply $a$ by itself $b$ times, ...
6
votes
5answers
304 views

Solve $4 \times2^x+3^x=5^x$ without any sort of calculator

Is there a way i can solve the following equation only by using high school mathematics? $$4 \times2^x+3^x=5^x$$ I tried writing $5$ as $2+3$ but didn't get any result. After that i tried to divide ...
2
votes
5answers
84 views

When is $\sqrt{x/y^2}$ equal to $\sqrt{x}/y$?

The solution to the quadratics is given by $r = -\dfrac{b}{2a}\pm\sqrt{\dfrac{b^2-4ac}{4a^2}}$, which is shortened to $r = -\dfrac{b}{2a}\pm\dfrac{\sqrt{b^2-4ac}}{2a}$, but I'm wondering how if this ...
-1
votes
2answers
43 views

$((-a)^3)^{1/2}\ne(-a)^{3*1/2}?$

The original problem is $\sqrt{(-a)^{3}}\sqrt{(-a)}$ I attempted to solve this as the following way: $\sqrt{(-a)^{3}}\sqrt{(-a)} = (-a)^{\frac{3}{2}}(-a)^{\frac{1}{2}}=(-a)^{2}=a^{2}$ However, I ...
7
votes
2answers
270 views

Fermat's Little Theorem and Euler's Theorem

I'm having trouble understanding clever applications of Fermat's Little Theorem and its generalization, Euler's Theorem. I already understand the derivation of both, but I can't think of ways to use ...
1
vote
4answers
69 views

Why is $-32^{\frac{1}{5}} = 2$

When you factorize $-32$, you get: $-32 = (-16) \cdot 2$ $-16 = (-8) \cdot 2$ $-8 = (-4) \cdot 2$ $-4 = (-2) \cdot 2$ $-32^{\frac{1}{5}} = -2$ The reason I am asking is because you get $-4 = -2 \...
1
vote
0answers
62 views

How does $3n+1$ change the proximity of $n$ to a power of two?

This is part of an attempt to prove Collatz's conjecture. I proved a modification of Collatz's conjecture, where instead of $3n+1$ if $n$ is odd, you do $n+1$. In Collatz's conjecture, if you get to a ...
-3
votes
2answers
67 views

How to solve $x^3+y^3=z^4$? [on hold]

Any three positive integers that are greater than 2 satisfying $x^3+y^3=z^4$ . You can not have two or more of the same number.
3
votes
1answer
123 views

Meaning of imaginary part of $\int_0^6 e^{x^3} dx$

My question is the title itself. How can it be possible that integral of real numbers can have an imaginary part?
-3
votes
2answers
39 views

What mean $x^\frac{a}{b}$ where $a,b \in\mathbb{R}$? [closed]

What mean $x^\frac{a}{b}$ where $a,b \in\mathbb{R}$? a) $\sqrt[b] {x^a}$ b) ($\sqrt[b] x)^a$ Thanks to all.
4
votes
4answers
204 views

Why is $\lim\limits_{x\to\infty} e^{\ln(y)} = e^{\,\lim\limits_{x\to\infty} \ln(y)}$?

In the above limit $y = x ^{\frac 1x}$. Is the above a limit or an exponent property? Thanks in advance. Context (Last paragraph): http://tutorial.math.lamar.edu/Classes/CalcI/LHospitalsRule.aspx
2
votes
4answers
173 views

“Exponential Madness” (Gauss's challenge)

From Euler's identity, we see that $e^{i\pi}=-1$ $\Rightarrow e^{2ik\pi}=1$ [squaring both sides]. This equation surely holds for all integers $k$. EDIT: From the second equation we get $e^{1+...
2
votes
1answer
683 views

Why does $\frac{1}{4}x^{-3/4} = \frac{1}{4x^{3/4}} = \frac{1}{4\sqrt[4]{x^3}}$?

This is taken from Khan Academy, I don't understand how these equate: $$\frac{1}{4}x^{-3/4} = \frac{1}{4x^{3/4}} = \frac{1}{4\sqrt[4]{x^3}}$$ How come the minus was remove from the original exponent?...
1
vote
2answers
25 views

Is there an imaginary Exponent with $a^x=-b$ while a,b>0

Is there an imaginary Exponent $x$ with $a^x=-b$ while a,b>0 ? An where could something like this be used?
0
votes
2answers
49 views

$e^{e^{10^{10^{2.8}}}}$ changing $e$ with $10$

From Numberphile $$e^{e^{10^{10^{2.8}}}}$$ changing $e$ with $10$, is there a way to change only the top most number while keeping all other numbers 10? i.e what is x in : $$e^{e^{10^{10^{2.8}}}} = ...
0
votes
1answer
34 views

Calculating the last digit of exponent

I need to calculate the last digit of $723^n$.(For every positive integer $n$). If it was to calculate the last digit of $a^b$ when I know the value of $a$ and $b$,then it was easy- for example,If I ...
2
votes
2answers
413 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 ...
4
votes
5answers
172 views

Find $3^{333} + 7^{777}\pmod{ 50}$

As title say, I need to find remainder of these to numbers. I know that here is plenty of similar questions, but non of these gives me right explanation. I always get stuck at some point (mostly right ...
9
votes
5answers
289 views

Which number is greater, $11^{11}$ or $9^{12}$?

Which number is greater than $11^{11}$ or $9^{12}$? My work so far: $11^{11}=285311670611>9^{12}=282429536481$. But to verify the validity of equality should be in the range of easily ...
2
votes
4answers
53 views

Derivative of exponent

Looking to solve : $$ \frac{d}{dx}[2^{0.5x}]$$ The multiplication and X value in the exponent is confusing me. Help? Thanks!
3
votes
8answers
286 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: ...
0
votes
0answers
36 views

question concerning tetration to infinity.

So I read on-line that \begin{equation} x^{x^{x^{x^{x^{x}}}}}=2 \end{equation} where number of x goes to infinity can be solved by solving \begin{equation} x^{2}=2 \end{equation} so by the same logic \...
3
votes
4answers
72 views

Why is the inverse tangent function not equivalent to the reciprocal of the tangent function?

I know that $$ {\tan}^2\theta = {\tan}\theta \cdot {\tan}\theta $$ So I guess the superscript on a trigonometric function is just like a normal superscript: $$ {\tan}^x\theta = {({\tan}\theta)}^{x} ...
1
vote
1answer
33 views

Seeming Contradiction With Complex Exponents

I was fooling around while waiting for a page to load and came across the following "contradiction". Let $x=(-1)^{i}$. Then $x^{i}=(-1)^{i\cdot i}=(-1)^{-1}=-1$. Thus, $x=\left(x^{i}\right)^{-i}=(-1)^...
0
votes
2answers
20 views

Strong Induction issue

I am trying to prove a statement using strong induction but I seem to be getting stuck. I don't know if did something wrong or I am just not recognizing an opportunity for factoring/how to factor ...
-1
votes
1answer
37 views

How do I get negative exponent? [closed]

I am trying to get negative exponent expression to work. Say I have 1.3 * 10^-3 I tried putting that between \$$ pair and got the following $$1.3 * 10^-3$$ I see 10-3 instead of seeing -3 in ...
2
votes
2answers
41 views

Logarithm of >2 numbers

I am learning logarithms and i found that $log(a*b) = log(a)+log(b)$ I tried to apply the same principle for three numbers like $log(a*b*c) = log(a)+log(b)+log(c)$ but it didn't work as i expected. ...
3
votes
4answers
127 views

Easy exponents question

I have the GRE Friday... I got hung up on this easy exponents problem (I think it was these exponents, don't recall exactly) $$\frac{6^{14}}{2^7 * 3^5} = ? $$ The answer is $(2^7)(3^9)$.
1
vote
1answer
75 views

Maybe this inequality holds? $x!-y!>x^n$?

Let $x,y,n$ be postive integers such that $x\ge 2y,y>n,n>3$ I conjectured that $$\color{red}{x!-y!\ge x^n}$$ Now, I claim that $$\color{red}{x!-y!=y![x(x-1)(x-2)\cdots(y+1)-1]\ge (n+1)![x(x-1)(...
11
votes
9answers
2k views

Which of the numbers is larger: $7^{94}$ or $9^{91} $?

In this problem, I guess b is larger, but not know how to prove it without going to lengthy calculations. It is highly appreciated if anyone can give me a help. Which number is larger $$\...
0
votes
1answer
92 views

What is the value of $e^{-10000}$?

What is the value of $e^{-10000}$? We know that the function $e$ does not attain value $0$ anymore. But in R and Matlab the value of $e^{-10000}$ is given as $0$ which is not correct anymore. I ...
0
votes
3answers
86 views

Simplify the expression $(a^2)^5(2x^2)^4\over2^5(a^3)^3(x^3)^2$

Can anybody please provide a step by step solution to the following expression ? $$(a^2)^5(2x^2)^4\over2^5(a^3)^3(x^3)^2$$
6
votes
7answers
300 views

Show that if $n>2$, then $(n!)^2>n^n$.

Show that if $n>2$, then $(n!)^2>n^n$. My work: I tried to apply induction. So, at the induction step, I need to prove, $n^n>(n+1)^{n-1}$ Here, I tried to use induction again without any ...
0
votes
0answers
48 views

Interpreting $\log_2\left(\frac{1}{0}\right)$ and $\log_2\left(\frac{0}{0}\right)$

I'm hoping someone can confirm what I've done is correct. I am working with biological datasets ... about 100 RNA-Seq datasets and I'm trying to analyze the relative up or down regulation of genes. ...
0
votes
3answers
50 views

Why is $\sqrt{x/x^{-1}}$ OR $\sqrt{x/{1/x}}$ = $\lvert x\rvert$ and not just x

I have this task: Find equal expression to square root of fraction of x and its inverted value (this is translated from my mother tongue so I'm sorry if I've used incorrect terms). Anyway the starting ...
2
votes
2answers
58k views

why does e raised to the power of negative infinity equal 0?

Why is it that e raised to the power of negative infinity would equal 0 instead of negative infinity? I am working on problems with regards to limits of integration, specifically improper integrals ...
0
votes
1answer
36 views

How to expand and solve using exponent laws and algebra?

How do you solve this question, specifically between lines 2 and 3 in the image? The intuition, expanding and math behind it would be much appreciated! Question needing solved
1
vote
2answers
53 views

Express $(100^3)^5$ with a base of $10$

Express $(100^3)^5$ with a base of $10$. I don't get this.
1
vote
2answers
46 views

How would one solve this exponent problem without a calculator?

This is a problem on a non-calculator portion of an algebra 2 honors worksheet. $$2^{24}=(3x-1)^8$$
2
votes
0answers
31 views

Exponential of a symmetric, tridiagonal matrix

Is there an analytic result for the exponential of a symmetric, tridiagonal matrix (the diagonal can be zero, if this helps). Moreover, if simplifies the result, the sub-diagonal and supra-diagonal (...
1
vote
1answer
92 views

Solve the system of equations $x^y=y^x$

Solve the system of equations $$ x^y=y^x \\ a^x=b^y $$ I could not solve this despite many tries
3
votes
4answers
388 views

Exponential Equations with Fractions

I have had some issues with the following two equations: $$ \frac{3^{n-2}}{9^{1-n}}=9$$ $$\frac{5^{3n-3}}{25^{n-3}}=125$$ If anyone could work them out step by step that would be awesome. I ...
-1
votes
2answers
26 views

Exponentiation - a to the i power

So if, $i=\sqrt{-1}$ and, $i=(-1)^\frac{1}{2}$ then, $a^i=a^{-1^\frac{1}{2}}=(\frac{1}{a})^{\frac{1}{2}}=\sqrt{\frac{1}{a}}$ ???
1
vote
0answers
13 views

The multivalued behaviour of complex exponential $z^\lambda$

On Gustav Doetch's Introduction to the Theory and Application of the Laplace Transform, it says: The power series $\sum_{n=0}^\infty a_nz^n$ converges on a circular disc. Replacing the integers $n$...
0
votes
1answer
24 views

Does equivariant property commute with matrix exponential?

Given a vector space $V$ and endomorphism $M : V \to V$ we can define the exponential of $M$: $\exp(M) = \sum_{k=0}^{\infty} \frac{1}{k!}M^k$. This has the property that it commutes with conjugation ...
9
votes
1answer
188 views

Solving an inequality : $n \geq 3$ , $n^{n} \lt (n!)^{2}$. [duplicate]

I proved this inequality in the following way: Lemma: $r \in \Bbb N, r \geq 3$. We have $r^r \gt (r+1)^{r-1}$. Proof: We apply the AM-GM inequality to the $r$ positive integers where there are $r-1$...
5
votes
5answers
106 views

For integer $n>2$, $(n!)^2 > n^n$ [duplicate]

Problem: For integer $n>2$, show that $(n!)^2 > n^n$ My attempt: I tried using induction. For $n=3$, the given condition is satisfied. Let us suppose $k!^2>k^k$ for some $k\geq3$. Then, $(...