Questions tagged [exponentiation]

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

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1answer
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How to convert any base 10 integer to a rational power of 2?

For example, through (lots of) trial and error with a calculator, I figured out that: 1.551e+25 is approximately equal to 2^83.6814 Is there an equation or algorithm for a conversion like this? ...
0
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5answers
62 views

Hard exponential equation

Is there any algebraic way of solving the following equation for $x$? $$\frac{3^x+2^x}{3^x-2^x}=7$$ Apparently there is some way of solving this and I heve tried to solve it in a conventional ...
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5answers
80 views

Is it true that $n \leq 2^{n-1}$ for all natural numbers $n > 0$? [closed]

Seems to be true but I want to make sure: $n \leq 2^{n-1}, n > 0$
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4answers
86 views

What is the answer to $17^{16} \pmod {10}$? Is it equal to $ 9 $ or $1$

I encounter a modular arithmetic problem. which says: Find the last Digit of $17^{16}$, by intuition the last digit of a number is the remainder of the number divided by $10$. so the statement is: $...
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3answers
138 views

Why is $49^{-\frac{1}{2}}=\frac{1}{7}$?

Why is $49^{-\frac{1}{2}}=\frac{1}{7}$? I know that $\sqrt[n]{m^p}=m^{\frac{p}{n}}$ so I figured I can state the above is $-\sqrt{49}=-7$, but that is incorrect. I can't put the negative inside the ...
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3answers
62 views

$(a^b)^c$ and $a^{(bc)}$ for complex numbers

I've just stumbled across $$\left(e^{2\pi i}\right)^{\frac{1}{2}}=\sqrt 1=1\neq -1=e^{\pi i},$$do I have some error in my thoughts there or does $(a^b)^c=a^{(bc)}$ not hold for complex numbers?
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1answer
37 views

Proof of equivalence between two methods of binary to decimal conversion.

I have two binary to decimal conversion methods and want a proof - or an intuition at least - of why they are equivalent. The first method is quite intuitive to me and seems to be more popular: $[...
8
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1answer
93 views

For every $b$ in the power $a^{b}$, does there exist an $a$ such that the digit sum of this power is equal to $a$?

$1^0 = 1\to 1 =1$ $x^1=x\to x=x\;\forall x$. $9^2 = 81\to 8+1=9$ $8^3=512\to 5+1+2=8$. $7^4=2401\to 2+4+0+1=7$ $46^5 = 205962976\to 2+0+5+9+6+2+9+7+6=46$ $64^6 = 68719476736\to 6+8+7+1+9+4+7+6+7+...
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2answers
26 views

How can I prove that $10=2^{a}*3^{b}*7^{c}$ has infinite solutions?

Both in a unrescrited case and with the following restriction: $a+b+c=1$
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2answers
62 views

How do I solve the for the base of an exponential modular arithmetic equation? [closed]

The question is: $$10 \equiv M^5 \mod{35}$$ How do I isolate and solve for $M$?
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2answers
77 views

multiplication of exponentials for non commuting matrices

Is there any special condition for the following statement to be true for any $n \times n$ matrices? $$ e^Ae^B=e^{A+B}=e^Be^A $$ Is it always correct to say that the exponential product equals ...
3
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1answer
155 views

What's the derivative of $\underbrace{x^{x^{x^{x^…}}}}_n$? Are my calculations right?

The problem: To make things easier on us so we don't have to use an underbrace, I'm going to declare: $$\Psi(x,n) := \underbrace{x^{x^{x^{x^{\dots}}}}}_n$$ And we evaluate $f$ from the topright to ...
0
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1answer
23 views

How can you express radicals as multiplication/addition?

How can you express radicals as multiplication/addition? Most mathematical operations clearly reduce to multiplication/addition (same thing), but how do you do that for exponentials/radicals? Thank ...
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1answer
11 views

Show that the following are equivalent (exponent laws)

Determine whether the two expressions are equivalent I was able to solve this by distributing the -2(1+x$^2$) and setting it equal to the expression x$^2$-4 and getting x$^2$=2/3 and then ...
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1answer
39 views

How to prove this theorem for exponential equations

Prove that if $a^x=a^y a∈R$ then $x=y$ as in the exponential equation $2^x=2^3$ How can I prove this theorem, if you know what I mean
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1answer
23 views

Modular exponentiation with the Carmichael function

This is something I have been thinking of using in a math competition against other players so it would be very helpful to me if it was explained. How would someone reduce a problem such as $\frac{7^{...
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2answers
66 views

How to decelerate from velocity v to stop time t over distance d?

I'd be grateful for some help with this problem I am trying to solve. Let's say that I have an object travelling at a velocity v. I want that object to come to a halt in time t AND travel exactly ...
0
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1answer
75 views

What is the difference between $\dfrac{1}{3^{-2}}$ and $3^{-2}$

Stupid question but how in the world does 1/3^-2 not be the same thing as 3^-2? The first answer I got is 9, the second one I got is 0.111... Isn't the negative power just 1/3/3? What difference does ...
2
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2answers
78 views

Why do we take $(1+z)^{\alpha}$ as $e^{\alpha \operatorname{Log}(1+z)}$?

Let $\alpha$ be a complex number. Show that if $(1+z)^{\alpha}$ is taken as $e^{\alpha \operatorname{Log}(1+z)}$, then for $|z|< 1$ \begin{equation*} (1+z)^{\alpha} = 1 + \frac{\alpha}{1}z + \...
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1answer
36 views

Near integers in powers of binomials with radicals

This question comes out of a mathematics calendar problem that asked for the tenths digit of the expression $(17 + \sqrt{280})^{17}$. The calendar implied the digit should be 9, but after playing ...
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2answers
31 views

Exponent Problem. Verify my solution.

Is this solution correct ? please click here to see the question & my answer Sorry for the nasty handwriting (in image). $$\sqrt{a}(2a^2-4/a)\Rightarrow a^{1/2}(2a^2-4a^{-1})\Rightarrow 2(a^{5/2}-...
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3answers
27 views

Special cases of the Laws of Exponents?

I have a few questions regarding powers. We have: \begin{align} (x\cdot y)^n &= x^n\cdot y^n \tag 1\\ (x^m)^n &= x^{m\cdot n} \tag 2\\ x^m \cdot x^n &= x^{m+n} \tag 3 \end{align} ...
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0answers
28 views

Prove that every number to a power greater than 1 can be written as $X^{n+1} = \Sigma_{i=1}^{n}(a_{i}X^{i}-b_{i}Y^{i})$

Prove that every number to a power greater than 1 can be written as $X^{n+1} = \Sigma_{i=1}^{n}(a_{i}X^{i}-b_{i}Y^{i})$ for all natural numbers $X,Y$ where $X>Y$. Further, there exist integers $...
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4answers
82 views

How to calculate $n$ in $f = p^n - q^n$?

I have the formula: $f(n) = \frac{p^n - q^n}{\sqrt 5}$ Assuming I know the value of $f(n)$, can I calculate $n$? Sure, I can convert to $ \sqrt 5*f(n) = p^n - q^n$ . But after that I stuck... edit ...
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1answer
17 views

Optimum way of self replication?

Asume you can build a self replicating machine called X. X can perform two functions, further build another machine X in 24 hours or generate a product Y, also in 24 hours. In a 30 day, calculate the ...
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0answers
23 views

Novice question about power laws

Say I have a cumulative distribution that is described by a power law, $$N = ax^b$$ I.e., N(x) is the number of objects greater or equal than x and $b<0$ so that there are more objects with small ...
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2answers
24 views

0 to the power of any number

I have proof that $0^n$ = undefined. Since, $2^5 = 32$, $2^4 = 16$, $2^5/2 = 32/2 = 16 = 2^4$. Similarly if $0^n = 0$. Then, $0^{n-1} = 0$ $0^0/0 = 0/0 = 0^{n-1}$. But $0/0$ is undefined. ...
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1answer
82 views

Find $f^{−100}g^{146}f^{301}$ (permutations to high powers)

Find $f^{-100}g^{146}f^{301}$ where $$f = \begin{pmatrix} 1 & 2 & 3& 4 & 5 & 6 & 7 \\ 3 & 1 & 5 & 7 & 2 & 6 & 4\end{pmatrix}, \\ g = \begin{pmatrix} 1 &...
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1answer
32 views

How to show this rule works for whole numbers to the fifth power?

Today I was shown a rule about natural numbers raised to the fifth power and an interesting method to generate them through the odd numbers. Start with $1 = 1^5$. Then skip the next $T_1 = 1$ odd ...
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3answers
43 views

How do I express $y = 1 - e^{-ax^b}$ as a linear function in terms of $y$, $x$, $a$, and $b$?

I tried using the log rules and eventually got to $\ln(\ln(1-y)) = \ln(-a) + b\ln(x)$, but $\ln(-a)$ is obviously not defined across all $a$. This assumes that $x\ge 0$ and that $y<1$.
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0answers
20 views

Cube pattern of last few digits [duplicate]

Consider the sequence $$ t_0 = 3 \qquad t_1 = 3^3 \qquad t_2 = 3^{3^3} \qquad t_3 = 3^{3^{3^3}} $$ defined by $$ t_0 = 3 \qquad \text{and} \qquad t_{n+1} = 3^{t_n}, \qquad n >=0 $$ What are ...
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2answers
34 views

Determine all real (a,b,c) that satisfy the systems of equations.

Determine all real (a,b,c) that satisfy the equations $a^2b^3c=-(2^7)$ $ab^2c^3=(2^8)$ $a^3bc^2=-(2^6)$ I have tried making every variable a base 2 (and (-2)) to an exponent and then use systems ...
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2answers
56 views

Properties and applications of complex logarithms and exponentials.

We have these two identities where $a>0$ real, $n$ integer and $z$ complex : \begin{align} (a^z)^n=a^{nz} \tag{1} \\ (a^n)^z=a^{nz} \tag{2} \end{align} $(1)$ is true. Is $(2)$ also true? ...
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2answers
37 views

Addition of different bases raised to the same power (solve for power). Is power determined uniquely?

Let a vector $\mathbf{x} = [x_1, \ x_2, \ \dots \ , \ x_n]' \in \mathbb{R}^n$ contain non-negative elements i.e. zeros and positive values. Problem: $x_1^a + x_2^a + \dots + x_n^a = c \quad$ ...
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4answers
39 views

What happens when a is negative in $ a^{\frac{m}{n}} $.

What happens when a is negative in $ a^{\frac{m}{n}} $. This question may seem silly but I got confused by it. If you take the example $2^{\frac{4}{3}}$ Method 1: $\sqrt[3]{2^4} \approx 2.51984$ ...
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3answers
86 views

What went wrong in proving $i=1$ [duplicate]

I started with $$x=(-16)^{\frac{1}{2}}$$ $$x=(-16)^{\frac{2}{4}}$$ Since $$(a^m)^n=a^{mn}$$ we have: $$x=((-16)^2)^{\frac{1}{4}}$$ $$x=((16^2)^{\frac{1}{4}}$$ $$x=\sqrt{16}=4$$ Hence $$(-16)^{\...
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1answer
34 views

Highest power of 4 that divides n

Let $n\in\mathbb{N}$ or $\mathbb{Z^+}$. I know there are some ways to compute/find the highest power of $2$ that divides $n$, there's the ruler sequence, and some ways to do this with boolean algebra ...
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2answers
27 views

Sum of n with factoral exponent

I'm not sure what this is called, but the application is in counting the number of nodes in a consistently branching structure. For example, $5$ nodes branch into $5$ nodes each, each branching again,...
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2answers
47 views

How to prove “if $2^x=2^y$ so $x=y$”

I got this to prove and I don't know how. I tried everything but I just don't know how to prove it. thanks for helping.
4
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3answers
93 views

Solving $\log_6(2x-3)+\log_6(x+5)=\log_3x$

Solve for $x$ I have an equation that I have been working on solving; I know the solution, but I cannot get to it myself. Almost every simplification I do reverts back to a previous step. Can anyone ...
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3answers
70 views

What does the notation $A^{-2}$ mean if $A$ is a matrix?

$A^2$ means to multiple the matrix by itself, and $A^{-1}$ refers to the matrix's inverse. Would $A^{-2}$ be the square of the inverse or the inverse of the square?
4
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4answers
146 views

Why is it that (negative number) ^ (irrational number) is nonreal?

We know that $(-1)^{\frac 12}$, $(-1)^{\frac 14}$, ... $(-1)^{\frac 1{even}}$ all result in complex answers, whereas $(-1)^{\frac 13}$, $(-1)^{\frac 15}$, ... $(-1)^{\frac 1{odd}}$ all result in real ...
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2answers
28 views

How to represent the given series in exponential form

I was working on a problem that devolved to the summation $\Sigma_{n=1}^\infty n \dfrac{x^{n-1}}{(n-1)!}$. Is there any exponential representation available for such a term?
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1answer
75 views

Use Euler's Theorem to reduce the power factor $P$ to a value $R$ less than $\phi(m).$

I need to calculate $12345^P \pmod m$, where $P=3^{124}+2$ and $m=53892647$. The only thing that is confusing me is that to reduce it using Euler's theorem $m$ needs to be square-free number however $...
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2answers
478 views

$-1$ to the power of a irrational number

According to Wolfram Alpha, $(-1)^\pi \approx -0.90 + 0.43i$. But $\pi$ has proven to be irrational (we can't write $\pi$ in terms of a fraction $a/b$ with $a$ and $b$ integers) and, in the real ...
2
votes
2answers
39 views

Proving a function is monotone

Let $n\in \mathbb{N}$, $u_1,u_2,\ldots ,u_n>0$ and I want to prove that the function $$p(\alpha)=\frac{\sum_{i=1}^n u_i^\alpha}{\left( \prod_{i=1}^n u_i^\alpha \right)^{1/n}}$$ is monotone in ...
0
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0answers
16 views

Power sum refinement inequality

We start with the following inequality : Let $a,b,c,d>0$ such that $a+b+c+d=4$ $$\sum_{cyc}a^{ab}\geq (Q)e^{\frac{P}{3(Q)}}>\pi$$ With : $$P=\sum_{cyc}a^{ab0.75}\ln(a^{ab0.75})$$ And $$Q=\...
6
votes
1answer
162 views

Prove that $x^{y^x} > y^{x^y}$ for $x > y > 1$

Prove that $x^{y^x} > y^{x^y}$ for $x > y > 1$. So I've tried this so far: $x^{y^x} > y^{x^y}$ $e^{(y^x)\ln x} > e^{(x^y)\ln y}$ $(y^x)\ln x > (x^y)\ln y$ $e^{\ln({...
0
votes
1answer
24 views

Property of Jordan Blocks?

Not sure how to write a matrix in TeX, but suppose I have a canonical matrix of Jordan blocks along its diagonal. If I take the exponential of this matrix B with $e^{Bt}$, is the resulting matrix just ...
1
vote
2answers
73 views

Show that you can calculate x ^ 62 using only 8 multiplications

I have this method: ...