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

learn more… | top users | synonyms (1)

1
vote
2answers
46 views

Question on power, If 2x^2x^2x^2x… =4 Solve for x

I've seen this random example, in which can anyone give me clue how to solve for $ x $ here?
0
votes
0answers
65 views

Inverse and named fixed values, with ↑↑?

The inverse of $+$ is $-$, of $\times$ is $/$ and of $\text{^}$ is Log. Continuing upwards hyperoperationally, what is the inverse of $↑↑$? Whats more somtimes values that are fixed are given ...
22
votes
7answers
3k views

Is $\exp(x)$ the same as $e^x$?

For homework I have to find the derivative of $\text {exp}(6x^5+4x^3)$ but I am not sure if this is equivalent to $e^{6x^5+4x^3}$ If there is a difference, what do I do to calculate the derivative of ...
1
vote
1answer
136 views

How to minimize $a \times b$ where $a^b≥x$?

For example, if $x$ is 1 billion, the smallest possible $a \times b$ will be $3 \times 19 = 57$. This is because: $2^{30} \ge 1000000000$ $2 \times 30 = 60 $ $3^{19} \ge 1000000000$ $3 \times 19 ...
3
votes
2answers
258 views

Is there an operation that takes $a^b$ and $a^c$, and returns $a^{bc}$?

I know that multiplying exponents of the same base will give you that base to the power of the sum of the exponents ($a^b \times a^c = a^{b+c}$), but is there anything that can be done with exponents ...
0
votes
6answers
95 views

Why does $(2^{20}+2^{20}+2^{20}+2^{21})=5\cdot 2^{20}$?

I did this question on artofproblemsolving.com and I do not understand the solution. Why do I have $5 \cdot 2^{20}$? Can anyone explain?
5
votes
1answer
135 views

For each irrational number $b$, does there exist an irrational number $a$ such that $a^b$ is rational?

It is well known that there exist two irrational numbers $a$ and $b$ such that $a^b$ is rational. By the way, I've been interested in the following two propositions. Proposition 1 : For each ...
-2
votes
1answer
85 views

What is 0 raised to 0 ???!!!! [duplicate]

I have read many articles on this confusion but i am still confused... My simple question is - What is $0^0$? What is the present agreement to this? I feel that it should be 1 as anything to ...
1
vote
4answers
57 views

Is the sum of two exponential function can be equivalent to a third exponential function? [closed]

What will be the sum of two exponential functions $2\exp(4 x) + 3 \exp(5 x)$ equivalent to a third exponential function? Is it possible?
1
vote
3answers
36 views

How to solve for f?

The question asks to solve for the variable: $$2=6(3^{4f-2})$$ I am not quite sure how to solve for $f$ because the bases on either side cannot be made equal. Here is an example of a similar ...
0
votes
1answer
43 views

Exponentiation for hash function & associativity

Some cryptographic papers use $H^n(x)$ to mean $H(H^{n-1}(x))$ where $H^0(x) = x$ and $H$ is a cryptographic hash. So $H^3(x)$ would be $H(H(H(x)))$. Is this definition formally correct? It seems to ...
1
vote
0answers
15 views

Evaluating Expressions with Integer Exponents

Simplify the expression by writing as a single power and then evaluate for a=-1,b=-2 and c=3. $$(b^{3}a^{4})^{2} \times (a^{3}c)^{3} \over ac^{3}$$ Here is what I did: ...
1
vote
1answer
30 views

Exponents with base close to $1$.

I was just fiddling around with a calculator and calculating powers of numbers really close to $1$ like $1.01,1.001\dots$ trying to find at what value they exceed $2$. This got me thinking if I could ...
0
votes
4answers
89 views

What is the reciprocal of $(-1/2)^k$?

What is the reciprocal of $(-1/2)^k$? The answer is meant to be $2^{-k}$ as if you flip something upside down the power becomes negative. However, I am not sure what happens to the negative in front ...
0
votes
1answer
36 views

How to compute $(a+1)^b\pmod{n}$ using $a^b\pmod{n}$?

As we know, we can compute $a^b \pmod{n}$ efficiently using Right-to-left binary method Modular exponentiation. Assume b is a prime number . Can we compute directly $(a+1)^b\pmod{n}$ using ...
2
votes
1answer
36 views

Audio frequency increment yielding wrong results

I am writing something in the ChucK programming language, which is designed specifically for audio time functions (in this case, hertz). I'm having a really difficult time with a mathematical ...
5
votes
0answers
97 views

The smallest non-zero integer $c$ such that $\sum\limits_{n=1}^m\pm(x+n)^6 = c$?

We have the neat equalities, I. Group 1 For $k=2,3,4,5,\dots$ $$\sum_{n=1}^{2^k}\epsilon_n(x+n)^k = 2^{\frac{k(k-1)}{2}}k! = 4,\;48,\;1536,\;\color{brown}{122880},\dots$$ for appropriate ...
0
votes
1answer
51 views

Can I simplify $ \ln(A/B)+C$ any more?

This should be a rather simple problem however I am having difficulty getting this simplified. If I need to simplify the expression $$ \ln(A/B)+C$$ My first step is $$ A/B + e^c$$ However MATLAB and ...
0
votes
4answers
58 views

How do you algebraically derive “x <= 0” from “-x = | x |”

A = "-x = | x |" B = "x <= 0" If A, then B. By plugging in numbers or testing ranges less than zero, greater than zero, and equal to zero, I can verify that A ...
4
votes
1answer
60 views

A problem with exponent laws…

Solve for $x$: $$x^{\frac 13}={32\over \sqrt{x}}$$ I'm not sure how start up this problem. I thought you had to multiply both sides by $\sqrt{x}$ so that it cancels out on the right side and moves ...
2
votes
0answers
36 views

How to reduce exponentiation expressions?

It is a simple question but I am afraid of its simplicity. Is that correct : $2^{30}+2^{30}+2^{30}+2^{30} = 2^{30}(1 + 1 + 1 + 1) = (2^{30})\cdot 4 = 2^{30}\cdot2^2 = 2^{32}$? I am doing complex ...
0
votes
2answers
28 views

Why is $-x = (x^2) ^ {\frac{1}{2}}$ not logically equivalent to $(-x) ^ 2 = ((x^2) ^ {\frac{1}{2}}) ^ 2$?

Why is $-x = (x^2) ^ {\frac{1}{2}}$ not logically equivalent to $(-x) ^ 2 = ((x^2) ^ {\frac{1}{2}}) ^ 2$ for all values of x? First equation: $-x = (x^2) ^ {\frac{1}{2}}$ Second equation: $(-x) ^ ...
1
vote
0answers
21 views

Help with repeated squaring

I'm having trouble figuring out how to use repeated squaring to figure out 289^377 mod 589. I've seen other websites break the exponent down into (1 + 4 + 16 ... ), but I'm not sure when to do that.
4
votes
0answers
59 views

Why can $x^0$ sometimes be simplified to 1 even when x can equal 0?

For example, the Taylor series for $e^x$ is $\sum_{n=0}^{\infty} \frac{x^n}{n!}$. It seems like it should be indeterminate or undefined at $x=0$, since the first term would contain $0^0$, but it's not ...
1
vote
0answers
38 views

Raising numbers with powers

So the question is as follows: Let $f(x) = \int_{0}^x \frac{x}{2\sqrt{t}}dt$. Suppose $f(f(f(...f(f(a))...)))$ (done $2013$ times) $= 2^{2013}$. Find the real-valued solution of $a$ Now, for my ...
2
votes
1answer
42 views

Pattern in Digits in Powers of 2

Along a similar line to this question, (pattern in decimal representation of powers of 5), I was playing around in a mathematics program called GAP. I was entering powers of two, when I noticed an ...
1
vote
1answer
44 views

How to simplify recurrence relation?

I'm having trouble seeing how $$5(2^{n-1} + 5\cdot 3^{n-1}) - 6(2^{n-2} + 5\cdot3^{n-2})$$ simplifies to: $$2^{n-2}\cdot (10 - 6) + 3^{n-2} \cdot (75 - 30)$$ How can I simplify the above ...
1
vote
1answer
25 views

Calculating the number of times a value must be halved for it to be less than or equal to another value

This is not a homework question; I'm working out an algorithm for an app I'm writing and I want to calculate the number of times I must halve a base value for it to be less than or equal to a minimum. ...
0
votes
1answer
60 views

Does (x^y)^a+b = x^(y*(a+b)) or x^((y*a)+b)?

I posses two Math books, both of which define a certain property of the algebraic manipulation of exponents in different ways. For example: Book one would claim that: 2^((3)2+3) = 2(3*5) = 2^15, ...
0
votes
1answer
61 views

An upper bound for $a_1 (\sum_{i=2}^n a_i^{n-1})$ in terms of $a_1^n + \sum_{i=2}^n a_i^n$

Assume that $n \in \mathbb{N}, n \geq 2$ and $a_i \in \mathbb{R}, a_i > 0, \forall i=1,...,n$ Show that $$ \frac{a_1 (\sum_{i=2}^n a_i^{n-1})}{ a_1^n + \sum_{i=2}^n a_i^n} \leq 1 -\frac{1}{n}$$ ...
2
votes
2answers
66 views

What is the correct value?

My confusion is: $(-9)^{2/3} = ((-9)^{2})^{1/3} = ((-9)^{(1/3)})^{2} = 4.32$ But my calculator shows math error, and google says: $(-9)^{2/3} = 2.16+3.74i$
3
votes
1answer
32 views

Simplifying exponents

I've been refreshing my maths over the last couple of weeks, and it's been a challenge since it has been a long time since I was actively using it (20+ years). Anyways, Khan Academy and old textbooks ...
5
votes
1answer
74 views

Is there a $k$ such that $2^n$ has $6$ as one of its digits for all $n\ge k$?

It is true that every power of $2$ of the form $2^{6+10x}$, $x\in\mathbb{N}$, has $6$ as one of its digits. Something more is true, the last two digits are either $64$ or $36$. The OP suggests that ...
0
votes
1answer
44 views

Logarithmic question

In the following question I fail to understand why the A option is correct. I understand that D is wrong, and that B and C are correct, but why is A correct? If $3^x=4^{x-1}$, then $x $cannot be ...
6
votes
2answers
374 views

Prove an inequality on natural number

Show that if $ a,b\in N$ and $a < b$, then $$\frac{a^a}{(a+1)^{a+1}} > \frac{b^b}{(b+1)^{b+1}}.$$
0
votes
1answer
21 views

Find $E[W|X>Y]$ where $W = X+Y$ and $X,Y \sim \exp(2)$ independently

I need an idea on how to solve following conditional expectation $E[W|X>Y]$ where $W = X+Y$ and $X,Y \sim \exp(2)$ and $X$ and $Y$ are independent. Thanks.
1
vote
1answer
33 views

How to prove that $f(x) = x^ε - \log x$ is $\infty$ when $x\to\infty$?

I'm trying to prove that the function $x^ε$ is "bigger" than $\log x$ when $x\to\infty$, for every $ε>0$. Or to put it in a more formal way: For every $ε>0$, there exists a constant $N$ for ...
0
votes
0answers
46 views

How to solve an equation with absolute value and x as exponent.

The inequation is: I thought that the solution would be the same as if there was no x as exponent, but in Microsoft Math it says it has no solution and about the equation it said it has solutions. ...
2
votes
1answer
24 views

Properties of the exponent function attached to a nonzero prime ideal in a Dedekind domain

I want to prove properties of $v_\mathfrak{p}$, which I have been told is: "the exponent function attached to a nonzero prime ideal $\mathfrak{p}$ that maps a given nonzero fractional ideal to the ...
2
votes
0answers
34 views

A relation with limits

Let $t \in \mathbb{R}_+$, $\varepsilon > 0$ and $p_\varepsilon \in [0, 1]$. There is in a book the following relation: $$ \underset{\varepsilon \rightarrow 0}{lim} (1 - p_\varepsilon)^{\lceil t / ...
-1
votes
1answer
4k 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 ...
1
vote
3answers
45 views

Operations and Identities [duplicate]

We have the binary operation addition on numbers. It has an additive identity ( 0 ) and it is commutative. Multiplication is simply repeated addition. It is a binary operation on numbers. Its ...
1
vote
4answers
47 views

How to see that $2^{n-1} + 2^{n-1} - 1 = 2^n - 1$

How to see that $2^{n-1} + 2^{n-1} - 1 = 2^n - 1$? Is there a rule about adding two powers of the same base I'm not aware of? I know that you can "add the exponents" if you are multiplying numbers ...
1
vote
1answer
21 views

Proof that. for any two natural numbers $n$, $m$, $n^m$ is not an $n$-ary pernicious number.

$a_{10}$ is defined to be an $n$-ary pernicious number when the digit sum of $a_n$ is prime in base $10$. How can I prove that, for any two natural numbers $n$, $m$, $n^m$ is not an $n$-ary ...
0
votes
2answers
34 views

Implicit logarithmic differentiation to find the horizontal tangents of an exponential function

The graph of $y = 6{(3{x}^2)}^x$ has two horizontal tangent lines. Find equations for both of them. $$ \\ \begin{align} \\ y &= 6{(3{x}^2)}^x \\ y &= 6 \cdot {3}^x \cdot {x}^{2x} \\ \ln{y} ...
0
votes
1answer
39 views

Variable exponent solve for x

How can I solve this exponent problem using simple math only? We need to solve for $x$ $2^{2x}-(3.2)^{x+2} + 32 = 0$ The second term here is $3.2$ not $3\cdot2$ ie 3 decimal 2 not 3 into 2. My ...
1
vote
0answers
29 views

Simplified Exponent in $\bmod$ equation

I am trying to simplified the following expression: $$\left(y^m\right)^{x} \bmod p$$ In my case, I can only solve $(y^x) \bmod p$ first without prior knowledge of $m$. Eventually, my answer should ...
0
votes
2answers
41 views

Is it wrong to “imagine” a one when thinking about exponentiation? (e.g. $3^2 = 1 \times 3 \times 3$)

This might be a bit of a basic question, but I'm going through Khan Academy to refresh my math skills in order to pursue a self-study of higher mathematics, so I'm really focused on the "why" of the ...
0
votes
1answer
23 views

Multiplication Question with Powers

A quick question. I dont know how to do this. $$1-2^k\times 2+1\times (-1)$$
0
votes
0answers
11 views

Find distortion exponent from Fourier fitting

I'm facing this problem in my master thesis: we are measuring the signal from a sensor which is, physically, a $\sin^2$ (or $\cos^2$). Some non idealities distort the signal by introducing an exponent ...