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

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3
votes
3answers
199 views

Why do real powers need positive bases? (and to handle them)

I'm studying Calculus and I've stumbled across a concept I have some difficulties in fully grasping. That is "real powers". I don't understand the theory behind it and I think I don't understand very ...
2
votes
2answers
287 views

Complex Exponent of Complex Numbers

How does one find the algebraic solution of a Complex number raised to the power of another Complex number? Here is the work I have done so far, if there are any mistakes please inform me. A real ...
22
votes
2answers
2k views

Rational number to the power of irrational number = irrational number. True?

I suggested the following problem to my friend: prove that there exist irrational numbers $a$ and $b$ such that $a^b$ is rational. The problem seems to have been discussed in this question. Now, his ...
1
vote
1answer
81 views

Positive rationals satisfying: $a^2+a^2=c^2$? [duplicate]

If there are none why not? Thanks in advance.
0
votes
2answers
93 views

How to calculate $2^{mn-1}/(2^n-1) \bmod{(10^9+7)}$

I was trying to solve Magical Five problem on codeforces. I have correctly formed an equation which I need to solve via program such that resulting number don't overflow. Answer can be Python or C++ ...
0
votes
0answers
57 views

Any irrational number can be raised to a power so that the result is an integer number [duplicate]

Does it hold in general, that for every irrational number there exists a power to which when raised, the result will be an integer? Does there exist a counterexample, of which it can be showed that no ...
2
votes
0answers
113 views

expansion of function on spherical harmonics

Let $f :S^{n-1}\longrightarrow R_+$ be a square integrable function. Let $\{Y_{j,k}\}$ be an orthonormal basis of spherical harmonics on $S^{n-1}$, $j\in Z_+$ and $k=1, \ldots, d_n(j)$, where $d_n(j)$ ...
0
votes
1answer
45 views

Does $\mathrm{Im}(\exp)$ being a manifold imply the domain is a manifold?

Let $G$ be a real matrix group of dimension $n$. Let $\mathfrak{g}$ denote the lie algebra of $G$. Suppose $X \subset \mathfrak{g}$ such that $e^X$ is a submanifold of $G$ of dimension $m$. Does ...
5
votes
2answers
482 views

Suggestions on how to prove the following equality. $a^{m+n}=a^m a^n$

Let $a$ be a nonzero number and $m$ and $n$ be integers. Prove the following equality: $a^{m+n}=a^{m}a^{n}$ I'm not really sure what direction to go in. I'm not sure if I need to show for $n$ ...
2
votes
3answers
855 views

Visual proof of the addition formula for $\sin^2(a+b)$?

Is there a visual proof of the addition formula for $\sin^2(a+b)$ ? The visual proof of the addition formula for $\sin(a+b)$ is here : Also it is easy to generalize (in any way: algebra , picture ...
2
votes
3answers
417 views

How to calculate the exponent of a given number.

Here's my problem : I can't figure how to get back my exponent Here the formula I use to get a given number : $$a(\text{const}) = 200,\quad b(\text{const}) = 1.1,\quad c(\text{var}) = 50$$ So $c$ ...
1
vote
1answer
411 views

zero raised to infinity

I encountered a question where the only condition stated that $t>0$ and was then asked to compare these two quantities $0^t$ $t^0$ The scope of $t$ is $(0,\infty)$ and hence for infinity 1.) ...
3
votes
3answers
83 views

Proving that not defined value is equal to something

My younger brother (9th Grader) got the following maths problem- Given: $$2^a = 3^b = 6^c$$ Prove: $$c=\frac{a * b}{a+b}$$ From my elementary knowledge of mathematics it seems like a=b=c=0.Also, ...
1
vote
3answers
141 views

Limit of ${x^{x^x}}$ as $x\to 0^+$

Can you please explain why \begin{align*} \lim_{x\to 0^+}{x^{x^x}} &= 0 \end{align*}
2
votes
1answer
102 views

Name of the $(-1)^n$ function?

Does the function $f\left(n\right)=\left(-1\right)^n, n \in \mathbb{Z}$ used in a lot of mathematical formulas have a special name ? EDIT: The context of this question is that I need a name for this ...
0
votes
3answers
101 views

Can $(-1)^{a+i b}$ be expressed without negative based exponentiation, complex exponentiation, complex logarithms or trigonometric functions?

Can this expression, where $a$ and $b$ are both real, be expressed without negative based exponentiation (i.e. $a^b$ where $a$ is negative), complex exponentiation, complex logarithms or trigonometric ...
4
votes
2answers
3k views

Proving the Product Rule for exponents with the same base

For all $ a, b \text{ and } c \in \mathbb{R}$ and $a>1$, Prove that $a^b\cdot a^c=a^{b+c}$ I have come across this question and its bugging me. Its a basic property that we learn in HS and I was ...
0
votes
1answer
49 views

Stuck on Elementary Exponentiation

I'm confused. Why is it that for a problem in the form of: $(2^{x+1})(2^{x-1})$ we get $2^{2x}$ instead of $4^{2x}$; Shouldn't we multiply the $2$s by each other..? Similarly for a problem like: ...
1
vote
2answers
80 views

How would I compute a fractional exponent, such as $\bigl(\frac{9}{16}\bigr)^{3/2}$?

How would I determine $\left(\frac{9}{16}\right)^{\frac{3}{2}}$? I've never encountered an exponent that was a fraction, or a least never without a calculator. What steps would I take to simplify ...
10
votes
1answer
465 views

Is 2201 really the only non-palindromic number whose cube is palindromic?

Hі, Wikipedia states that 2201 is the "only known non-palindromic number whose cube is palindromic", and lists no reference. It is in fact true that $2201^3=10662526601$, which is a palindrome. But ...
12
votes
2answers
1k views

Why are the first few powers of $2^{10}$ a little more than those of 1000?

See the complete list here: http://en.wikipedia.org/wiki/Power_of_two#Powers_of_1024. I'm wondering if there's a mathematical explanation for the relationship or if it's just coincidence.
3
votes
1answer
150 views

Is there a name for raising a number to it's own value as a power

Is there a mathematical term for raising a a number to the power of its own value.... eg. $5^5 , 1692^{1692}$. Many Thanks
11
votes
6answers
1k views

Matrix to power $2012$

How to calculate $A^{2012}$? $A = \left[\begin{array}{ccc}3&-1&-2\\2&0&-2\\2&-1&-1\end{array}\right]$ How can one calculate this? It must be tricky or something, cause there ...
2
votes
2answers
85 views

Large exponential modular

Proof $2011^{2011^{2011}}-2011 \equiv 0 \mod 30030$ By Chinese Remainder Theorem this is equivalent to proving: $2011^{2011^{2011}}-2011 \equiv 0 \mod 2$ $2011^{2011^{2011}}-2011 \equiv 0 \mod 3$ ...
0
votes
1answer
434 views

How to check if Mersenne number is prime?

How can I prove that that Mersenne number, number of the type $2^n - 1$ is prime number. One theorem says that if $2^n - 1$ is prime than $n$ is prime number also. But this doesn't work vice versa. ...
4
votes
3answers
225 views

Can we *ever* use certain log/exp identities in the complex case?

This article on Wikipedia points out that certain identities for the log and exponential functions which are familiar from the real case require care when used in the complex case. Failures in the ...
2
votes
1answer
77 views

Why inverse modulo exponentiation is harder than inverse exponentiation without modulo

I am new to number theory. I read in cryptography inverse modulo exponentiation is used because it is hard. But I couldn't understand the advantage of it over inverse exponentiation without modulo. ...
5
votes
3answers
442 views

Solve an equation with $e^{(x-2)}=e^{4}\cdot e^{\sqrt{x}}$

$$e^{(x-2)}=e^{4}e^\sqrt{x}$$ I know that $x = 9$ and I can show the calculations like this: $$e^{(x-2)} = e^{\sqrt{x}+4}$$ and now I need to get the $x$ to the right side but I dont know how.
1
vote
1answer
76 views

How to calc $ a^{2^n}$ mod $m$ in less than O(n) time?

a,m are positive integers. if needed, you can assume m is a prime. Is there any fast algorithm? I'm sorry for my not clear description.
4
votes
2answers
121 views

Characterizing continuous exponential functions for a topological field

Given a topological field $K$ that admits a non-trivial continuous exponential function $E$, must every non-trivial continuous exponential function $E'$ on $K$ be of the form $E'(x)=E(r\sigma (x))$ ...
2
votes
3answers
107 views

Why does $(-2^2)^3$ equal $-64$ and not $64$?

The title says it all. Why does $(-2^2)^3$ equal $-64$ and not $64$? This was on my algebra final, and I am completely stuck on how it works.
6
votes
0answers
216 views

Close power towers

In another question I gave some ways to determine which of two power towers is larger, but my answer there is incomplete because it doesn't handle the case where two towers are very close at each ...
0
votes
2answers
77 views

Square roots and powers

This is a rather silly question. In what order does one evaluate a combination of powers and fractional powers? I have the question phrased: $\sqrt{ 1/4^2 }$ OR ${ 1/4 }$? Which is greater? I ...
2
votes
1answer
54 views

Derivative of $x\times n - 2^{\log_2 {x \times n}}$

I have a problem with solving derivative of $f(x)$ in this case: $$f(x) = x\times 10^9 - 2^{\log_{10} x\times 10^9}$$ This is what I have: $$f^\prime(x) = \lim_{m\to0} {f(x+m) - f(x)\over m}$$ $$= ...
1
vote
1answer
32 views

Using exponent laws to simplify $2^t\times4^2+8^{t-2}$

This is one of the questions in my maths text book and I don't quite get it, can someone please explain it and show the working out? use index laws to find $2^t\times4^2+8^{t-2}$ Observation ...
1
vote
3answers
155 views

solving the equation$(((e^x)^{e^x})^{e^x})^{…}=2$ for x

Let $ \left(x_n\right)_{n = 1}^\infty $ be a sequence of real numbers defined by the initial value $ x_1 = e^x $ for some $ x \in \mathbb{R} $ and the relationship $ x_{n+1} = e^{x^{x_n}} $, such ...
0
votes
3answers
172 views

$1500=P \times { (1 + 0.02) }^{ 24 }$, what is the value of $P$?

Hey guys could you please tell me what is the faster why to solve this equation. It's a compound interest equation and I'm stuck at the ${ (1 + 0.02) }^{ 24 }$ I really don't know how to proceed in ...
6
votes
3answers
221 views

If $\theta\in\mathbb{Q}$, is it true that $(\cos \theta + i \sin \theta)^\alpha = \cos(\alpha\theta) + i \sin(\alpha\theta)$?

Is the following true if $\theta\in\mathbb{Q}$? $$(\cos \theta + i \sin \theta)^\alpha = \cos(\alpha\theta) + i \sin(\alpha\theta)$$ Is it true if $\alpha\in\mathbb{R}$? In each case, prove or give a ...
1
vote
2answers
96 views

Do any issues arise if we try to raise an element of $\mathbb{R}^+$ to an element of $\mathbb{C}$?

If $a$ and $b$ are non-zero natural numbers, the definition of $a^b$ is clear. Now it seems to me that there are (at least) two distinct ways of generalizing to larger number systems. Firstly, given ...
1
vote
1answer
123 views

I just found out that $0^0$ equals $1$, why is this? [duplicate]

I have done a lot of math so far, but I never stumbled on something this simple and yet mind boggling. Can someone tell me why $0^0$ equals $1$? I always knew that everything raised to a power of $0$ ...
0
votes
1answer
151 views

k-fold matrix product

For $k \in \mathbb{N}$, $B,C \in \mathbb{R^{n,n}}$, given the matrices $B,C$ , calculate all powers $B^k$ and $C^k$ I'm a bit puzzled by this task. I assume it's supposed to practice handling ...
2
votes
1answer
116 views

Exponential equations involving natural numbers at power “x”

Find x : $$4^x+15^x=9^x+10^x(2^x-3^x)(2^x-3^x-5^x)$$
4
votes
1answer
291 views

Digits in a large power of two

I am trying to find the answer to: 2^34359738368. As to be expected every calculator and computer program I have used has crashed. To be honest I don't even want to know the exact answer, I just want ...
3
votes
2answers
305 views

Zeta function and probability

I know that $\zeta(n) = \displaystyle\sum_{k=1}^\infty \frac{1}{k^n}$ (Where $\zeta(n)$ is the Riemann zeta function) But the reciprocal of $\zeta(n)$ for $n$ a positive integer is equal to the ...
4
votes
2answers
132 views

Proof for $\displaystyle\sum_{k=1}^n k^a$ equaling a sum of fractions

I know $\displaystyle\sum_{k=1}^n k^2$ equals $n/6+n^2/2+n^3/3$, but... why? And I also know that $\displaystyle\sum_{k=1}^n k^3$ equals $n^2/4+n^3/2+n^4/4$, but... is there a pattern so I can easily ...
1
vote
2answers
125 views

Is $a^n$ an exponent of $a$?

I see that definition says that $n$ is an exponent. But, the name of the function is normally expanded to the results computed by the function. That is, if we raise $a$ into power $n$ we say that ...
2
votes
1answer
127 views

Proof of correctness of Putzers algorithm

I have a question regarding the proof (seen below) of Putzers algorithm for matrix exponentiation. It's written by our danish lecturer at the university, so I translated the important parts into ...
1
vote
3answers
190 views

Integrating $x^x$ and getting a graph [duplicate]

I've heard many times of functions that cannot be integrated. For example, $x^x$, which is the most common. But what I don't know is how could you, even if the graph has no equation, plot this ...
6
votes
6answers
326 views

How can $4^x = 4^{400}+4^{400}+4^{400}+4^{400}$ have the solution $x=401$?

How can $4^x = 4^{400} + 4^{400} + 4^{400} + 4^{400}$ have the solution $x = 401$? Can someone explain to me how this works in a simple way?
7
votes
2answers
163 views

Power tower inequality

I want to prove the following power tower inequality: $$ 3 \uparrow \uparrow 100 > 4 \uparrow \uparrow 99 $$ but I don't know how to do this. I think that induction will not work, because I think ...