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
36 views

Compare three numbers, expressed as powers

So I have these numbers: $4^{68}, 5^{51}, 12^{23}$ They need to be ordered from the smallest to greatest. I have no idea how to solve this. Typically, one should break down the exponents somehow to ...
1
vote
1answer
45 views

Solviang a Diophantine equation:$y^x=x^{2007}$($x$ and $y$ integers).

I found this Diophantine equation and to solve it I used the definition of logarithm but the solution doesn't need of the use of logarithmic rules. I solved it in this way: $$y^x=x^{2007}$$ ...
0
votes
3answers
14 views

Algebraic simplification of likelihood ratio

Can someone help me understand how this: ...
0
votes
2answers
10 views

Properties of exponents when dealing with induction.

This will most likely be a simple question for most of you. While watching my lecture today the white board cut out and the instructor didn't explain the final step in an example. He went from ...
0
votes
2answers
23 views

indices and powers

If $a^3 = b^2$, which of the following statements could be true? $a \lt 0$ and $b \gt 0$ $a \gt 0$ and $b \lt 0$ $a \gt 0$ and $b \gt 0$ Can anyone explain the answer with examples?
3
votes
0answers
17 views

Square Roots: Variables with Exponents.

Alright, so let me get this straight: $\sqrt{x^2} = |x|$ $\sqrt{x^3} = x\sqrt{x}$ $\sqrt{x^4} = x^2$ $\sqrt{x^6} = |x^3|$ Are these correct?
2
votes
3answers
50 views

Comparing the size of $(\sqrt{5})^e$ and $e^{\sqrt{5}}$

So I have to figure out which one is bigger between $(\sqrt{5})^e$ and $e^{\sqrt{5}}$. After some trial and error I've come to the conclusion that $(\sqrt{5})^e > e^{\sqrt{5}}$. But of course I ...
1
vote
0answers
40 views

Algorithm for tetration to work with floating point numbers

So far, I've figured out an algorithm for tetration that works. However, although the variable a can be floating or integer, unfortunately, the variable ...
3
votes
1answer
38 views

Why x by x is NOT equal to squared x within exponents?

Normally you can write $x*x=x^2$ But if you are operating within exponents, $a^{x*x} \neq a^{x^{2}}$ as the latter is equal to $a^{2x}$. Is it a problem of notation ? [Edited] Thank you to having ...
0
votes
0answers
25 views

Symbol for exponentiation of a sequence? (Equivalent to SIGMA for summation and PI for product)

I have a student asking whether there is a symbol for exponentiation of a sequence? So there's SIGMA for summation of a sequence, PI for multiplication of a sequence and perhaps something else for ...
-5
votes
2answers
43 views

What is e^e^x? Also, what is log e^e^x to the base e, i.e, ln(e^e^x)? [closed]

What is e^e^x? Also, what is log e^e^x to the base e, i.e ln(e^e^x)? Thank you.
3
votes
2answers
33 views

unsure how to rearrange $f(x)$ into suitable $p(x)/q(x)$

Consider the function $f(x)= (x^3 + 2x - 3) / (x^2 + 3x + 4)$ by dividing the numerator and denominator by the highest power of $x$ present, convert $f(x)$ into the form $P(x)/Q(x)$ where both $P(x)$ ...
2
votes
1answer
48 views

Is there any condition while applying law of exponents?

${[(-3)^2]}^\frac{1}{2}$ = ${(-3)^2}^\frac{1}{2}$ = $-3^1$ = $-3$ But counted other way it is $9^\frac{1}{2} = \surd{9} = 3$ where I went wrong?
6
votes
4answers
509 views

How do pocket calculators calculate exponents?

I'd like to know specifically how a pocket calculator (TI calculators also apply) calculates $e^{0.1}$, and what methods or algorithms pocket calculators use in order to produce their answer.
1
vote
1answer
34 views

Combining log terms

I have this particular problem. We have to combine the log terms into a single log term: $$\dfrac{(2\ln a- \ln b - 5\ln c)}{2}$$ I did it in the following way : $$''~= \ln a -\frac{1}{2}\ln b - ...
2
votes
2answers
43 views

When (and why) did the convention that exponents are evaluated from right to left arise?

Earlier, I saw this question on Quora: X^Y^Z Which one do I do first? and the current most-upvoted answer is this: The ^ operator is not associative, so that: (X^Y)^Z is not the same value as ...
1
vote
2answers
31 views

How many uranium-238 atoms are left after 1.338 x 10^10 years?

The half-life of uranium-238 is about 4.46 x 10^9 years. How many will there be after 1.338 x 10^10 years? How can I figure this out? I know it's exponential, but how?
1
vote
0answers
42 views

series of powers of integer powers

Given two real positive numbers $a,b\in(0,\infty)$ and a series of natural integers $n=1,2,3,\dots$, is there any known formula to apply in order to calculate the series $$s(n)=a^{b^n}?$$ My goal is ...
1
vote
2answers
34 views

Simplification of powers

I think this is a really simple question, but for some reason my brain can't get round it. I am proving a combinatorial result by probabilistic method and the last step has got me really confused. ...
1
vote
1answer
14 views

Confusion with repeated exponents

When someone writes: $3^{3^3}$ Do they mean $3^{(3^{3})}=3^{27}$ OR ${{(3^3)}^3} = 27^3$ ? There are no brackets Please reply ... this may be a silly question ... Thanks!
0
votes
1answer
33 views

Solving for single variable proving to be extremely difficult.

I have been at this equation for about two days now, and I can not for the life of me find a way to solve to i. If anyone can please show me a step by step into solving this, it would help me out so ...
0
votes
1answer
49 views

Example for $a^k\equiv b^k$ and $k\equiv j$ but $a^j\not\equiv b^j\pmod n$

I need some help in the number theory please , Who can give me an example : If $$a^k≡b^k \pmod{n}$$ and $$k≡j \pmod{n}$$ is not necessary to be $$a^j≡b^j \pmod{n}$$
1
vote
2answers
32 views

Gaussian distribution raised to a power

Given that $X$ follows a Gaussian distribution $e^{-x^2/2\sigma^2}$, what distribution is followed by $X^{1/3}$? How does one start to solve this problem? I guess it isn't ...
0
votes
0answers
17 views

Ask about description of exponent number.

I'm a Taiwanese student and face an English describe problem. My teacher told me " 1.34 x 10 of negative 4 exponent means 1.34 x 10^(-4)" in English description. And I wander why this exponent is use ...
0
votes
1answer
34 views

Factorals with exponents. Is their a way?

I know of multiplication factorials with the 4! = 4*3*2*1 and I know of the addition with the nth triangle. I am busy deriving my own equation for something, and i am getting stuck on how to furthur ...
1
vote
2answers
43 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
58 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 ...
-1
votes
2answers
59 views

Find remainder of $3^{12} + 5^{12}$ when divided by $13$ [closed]

What is the remainder when $3^{12} + 5^{12}$ is divided by $13$?
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 ...
2
votes
3answers
163 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
93 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
115 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
67 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
51 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
35 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
33 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
14 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
25 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
73 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
34 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
83 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
47 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
53 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
34 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
26 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
18 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
36 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 ...