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

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3
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1answer
53 views

What's the name for this mathematical device used by programmers?

So a friend is trying to figure out what this is called so we can read more about it. The concept is/was used by database designers, who needed a compact way to store a list of selected options as a ...
4
votes
1answer
26 views

Several values of irrational exponentiation

When talking about a number to a rational exponent, there are as many answers as the denominator of the exponent. Like the question: Is $9^{1/2}$ equal to $3$ or $-3$. However when we have an ...
1
vote
0answers
40 views

A simple question on the matrix exponential

Probably a trivial question. Given two random matrices $A, B$ such that $\left\langle \left[A,B\right]\right\rangle =0$, namely only the (element-wise) mean of the commutator is zero, can I say that ...
43
votes
11answers
7k views

What exactly IS a square root?

It's come to my attention that I don't actually understand what a square root really is (the operation). The only way I know of to take square roots (or nth root, for that matter) it to know the ...
4
votes
4answers
111 views

Prove that $a^x$ is continuous

I'm having trouble with proving the following: Let $a > 0$ be a positive real number. Show that the function $f : \mathbb{R} \rightarrow \mathbb{R}$ defined by $f(x) := a^x$ is continuous. I'm a ...
1
vote
3answers
83 views

Solve for $n$ in $2^n=8$

So, I was wondering if it is possible to solve for $n$ in $2^n=8$ (or any other question where $n$ is a power) using $9^{th}$ grade math. Please excuse my naïveté if this is extremely stupid/simple. ...
5
votes
4answers
707 views

Number raised to power of irrational number

What is the consequence of raising a number to the power of irrational number? Ex: $2^\pi , 5^\sqrt2$ Does this mathematically makes sense? (Are there any problems in physics world where we ...
0
votes
1answer
99 views

Inverse of $f(x) = a \left(1 + \frac{c}{(1+x^b)^{-\frac{1}{b}} - c}\right) \cdot (1+x^{-b})^\frac{1}{b}$?

How can one find the inverse of $$ f(x) = \mathrm{a} \left(1 + \frac{\mathrm{c}}{(1+x^\mathrm{b})^{-\frac{1}{\mathrm{b}}} - \mathrm{c}}\right) \cdot (1+x^{-\mathrm{b}})^\frac{1}{\mathrm{b}} $$ with ...
3
votes
4answers
232 views

Solve this logarithmic equation: $2^{2-\ln x}+2^{2+\ln x}=8$

Solve this logarithmic equation: $2^{2-\ln x}+2^{2+\ln x}=8$. I thought to write $$\dfrac{2^2}{2^{\ln(x)}} + 2^2 \cdot 2^{\ln(x)} = 2^3 \implies \dfrac{2^2 + 2^2 \cdot ...
0
votes
0answers
34 views

Is locally lipschitz the power function?

Definition:(Hale,1980) A function $f(x)$ defined in a domain $D$ in $R^{n}$ is said to be locally Lipschitzian in $x$ if for any closed bounded set $U$ in $D$ there is a $k=k_{U}$ such that ...
0
votes
1answer
34 views

Solve for $N$: $2000N=(0.9025)^{\log_2 N}$

I want to find the value of $N$ while $2000N=(0.9025)^{\log_2 N}$ (This is sample value not actual) How to solve it? The Whole Question which i am solving is $Pe=(Pt/N)(1-δ)^{\log_2 N}$ ...
7
votes
2answers
370 views

Continuum between addition, multiplication and exponentiation?

I noticed this old post which attempts to find the shades of grey between a linear and log scale where results are between zero and one. However, I was looking for the more general case where we find ...
1
vote
1answer
108 views

Two hard indices questions, (what is power to a power of fraction) and (how is $(2^x)^2 = 4^x $)

The answer for question 1) is $2^{3b+6}$ Question 2 I only don't get the $Y^2$ bit
0
votes
1answer
29 views

Why does the graph of an exponential function shoot straight up when getting to x=1 in an exponential growth function with x^huge number?

I used to notice that when x is raised to the power of a huge number, the graph shoots up at x=1. Why does this happen?
3
votes
1answer
51 views

Why an exponential graph can't have b equal to 1

I've seen that the graph of an exponential function, $f(x) = a\cdot b^x$, cannot have $b$ equal $1$. Why is this? I think it's because the function would be a flat line if $b=1$. Is this true?
1
vote
3answers
50 views

Any reason an exponential decay function approaches but doesn't cross the x-axis?

I've seen graphs of exponential decay functions (where a>0 and 0 is less than b is less than 1) and they don't seem to cross the x-axis. I think it's true. Any reason this happens?
2
votes
1answer
42 views

Confusion about exponents like ${x^m}^{(1/n)}$.

I've been reading this post. It says that $\sqrt[m]{x^n} = x^{n\frac 1m}=x^{\frac mn}=x$ if $m=n$. Let's take $x=-2$, and $m=n=2$. Now we have, $\sqrt[2]{(-2)^2}=\sqrt[2]{4}=2$ But according to that ...
5
votes
3answers
114 views

Does $1^i$ and $1^{\frac{0}{0}}$ also give $1$ again? [duplicate]

This is the property of Real number $1$ that, $1^n=1$ does this property only hold $\forall n \in \mathbb R$ or also $1^i=1$ and $1^{\frac{0}{0}}=1$ If it is; explain how? I think that it should ...
3
votes
4answers
98 views

Why is $n^0 = 1$? [duplicate]

Why is any number to the zeroeth power equal to 1? I would think it would be equal to zero, since nothing multiplied by nothing is, well, I would think 0. But it is 1? Examples: $(-5)^0 = 1$; $0^0 ...
1
vote
1answer
30 views

Exponent identities with imaginary exponents$\left(a^i\right)^i$

I've been trying to understand how imaginary exponents work, and I think I mostly understand it, but I'm confused by something like $\left(a^i\right)^i$ (where $a$ is real). According to the normal ...
5
votes
4answers
329 views

Compare three numbers, expressed as powers: $4^{68}$, $5^{51}$ and $12^{23}$

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
80 views

Solving 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 require the use of logarithmic rules. I solved it in this way: $$y^x=x^{2007}$$ ...
1
vote
3answers
27 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 ...
-1
votes
2answers
27 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?
4
votes
1answer
26 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
61 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 ...
5
votes
2answers
357 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
44 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
59 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 ...
3
votes
2answers
42 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
56 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?
7
votes
4answers
959 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
40 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
52 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
34 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
49 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
36 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
15 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
41 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
56 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
45 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
1answer
39 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 ...
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2answers
52 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?
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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
137 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
264 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
137 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 ...