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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3answers
88 views

Exponents with the same power

I've wanted to practice solving simple operations on exponents, so I've made a couple of equations to which I know the answers. $$5^x -4^x = 9$$ I feel really stupid, because I can't solve this one ...
1
vote
1answer
104 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 ...
0
votes
1answer
43 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 ...
2
votes
1answer
39 views

How do we derive the sum of $3^n$ and $2^n$

I know that $\quad\sum2^n = 2 (2^n-1)$ How can we derive this summation? And also how can we deduce the summation of $3^n$ from this ? I did observe this pattern : $$ \begin{align} n &= 1 ;\ ...
3
votes
0answers
61 views

Solution to following functional equation

Consider the functional equation problem $$ f: \Bbb{R} \rightarrow \Bbb{R}$$ $$ f(a^b) = f(a)^{f(b)},$$ when $a,b \in \Bbb{R}, a,b \ge 0.$ So far the only solution I have is the trivial $$ f(x) ...
-1
votes
1answer
41 views

Integrals of powers? [on hold]

I need an integral of the form $\int(f(x))^ndx$. I know about $x^n$ and $e^{nx}$ and other various things you can do with the both of them, but are there closed forms for any other sort of function ...
0
votes
1answer
16 views

Values of $a$ for which the equation $100^{-\lvert x \rvert} - x^2 = a^2$ has the maximum amount of solutions

$100^{-\lvert x \rvert} - x^2 = a^2$ I don't know how to approach this problem, due to the x in the exponent. I would appreciate hints more than outright solutions :)
11
votes
4answers
722 views

why is $2.2250738585072014\text{e}{-308}$ not a number? [closed]

In programming the min value of a float is: $$2.2250738585072014\text{e}{-308}$$ but when I type this into a calculator, it says Not a Number. what I am wondering ...
19
votes
7answers
28k views

How to calculate a decimal power of a number

I wish to calculate a power like $$2.14 ^ {2.14}$$ When I ask my calculator to do it, I just get an answer, but I want to see the calculation. So my question is, how to calculate this with a pen, ...
0
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0answers
23 views

Demonstration of exponentiation with induction

How can you demonstrate that $a^0 = 1$ and that $a^{-n} = (1/a)^n$ using the principle of mathematical induction?
0
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2answers
20 views

exponent problem solving

I came across a problem; $a^x=b^y=c^z$ and $b^2=ac$. It is required to show $\frac{1}{x}+\frac{1}{z}=\frac{2}{y}$. I have tried the following steps- \begin{equation*} b^2=ac \\ b=\sqrt{ac} \\ ...
1
vote
1answer
43 views

Can we write “fractional root” symbol in math?

Fractional exponents are legit but I have never seen fractional roots, so I just wonder if we can write fractional roots such as this: It sometimes can be convenient to think about too.
2
votes
1answer
59 views

Evaluate the limit $\lim_{n\to\infty} \frac{3^n}{2^n+3^n} $

It seems reasonable to assume that $$\lim_{n\to\infty} \frac{3^n}{2^n+3^n} $$ goes to zero but I can't figure out how to prove it.
1
vote
2answers
47 views

$\lim_{n \to \infty} \frac{(-2)^n+3^n}{(-2)^{n+1}+3^{n+1}}=?$

I have a question: $$\lim_{n \to \infty} \frac{(-2)^n+3^n}{(-2)^{n+1}+3^{n+1}}=?$$ Thanks for your help>
1
vote
4answers
131 views

Finding $\displaystyle \lim_{n \to \infty} \frac{2^{n+1} + 3^{n+1}}{2^{n}+3^{n}}$

I need help with finding $\displaystyle \lim_{n \to \infty} \frac{2^{n+1} + 3^{n+1}}{2^{n}+3^{n}}$ Thanks!
1
vote
1answer
36 views

How to solve for x in $2^{2x^2}+2^{x^2 + 2x + 2} =2^{5+4x}$

This is the question: $$\large{2^{2x^2}+2^{x^2 + 2x + 2} =2^{5+4x}}$$ What I did was put $~\large{2^{x^{2}}=t}$ From this, I got, roots of the quadratic: $$\large{-2^{x+1}\pm~\left( ...
0
votes
2answers
33 views

Continued addition and under rooting of 12

$\sqrt{(12 + \sqrt{12......})}$ and so on.... How do I find its answer? This is a question in our class VII mats book. P.S. - Answer is 4
0
votes
2answers
47 views

Square Root of $320$

Given, $$\sqrt{5} = 2.236$$ $$\sqrt{320} = 2^3 \times \sqrt{5} = 8 \times 2.236 = 17.888$$ This is the explanation provided in my school book. Could someone please elaborate ? Thanks in ...
4
votes
4answers
99 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 ...
2
votes
3answers
3k views

Raising a square matrix to a negative half power

I want to implement the following formula (taken from Kaiser, 1970) in R where $R$ is square matrix of correlations: $$S = (\textrm{diag } R^{-1})^{-1/2}$$ I understand the diagonal and inverse ...
5
votes
1answer
269 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 ...
0
votes
4answers
24 views

Why is 'something per hectare' denoted with a negative exponent ( $ha^{-1}$)?

Quick question.... why is it that something per hectare is shown as having a negative exponent, $ha^{-1}$? For example, on this page: http://www.ipcc.ch/ipccreports/sres/land_use/index.php?idp=12 1 ...
0
votes
3answers
54 views

Formula for exponents?

Is there a formula for exponents that works with both negative and positive powers? I have tried searching online but only found: If positive do this, if negative do this. Thanks. EDIT: Ah, I see ...
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 ...
1
vote
1answer
14 views

Finding a base given an exponent

In math, the logarithm of a number $n$ in base 10, finds the exponent where 10 has to be raised to, to produce $n$ again. So if $Log_{10}(n) = p$ then $10^p = n$. What I'm looking for is essentially ...
5
votes
3answers
54 views

Show $x^n \geq x_1^n+x_2^n+\ldots+x_k^n \Bigg\vert x_1+x_2+\ldots+x_k = x, x \geq 0, n \geq 1, k\in\mathbb{Z}$

Hello StackExchange Community, This is my first post on the forum. Please forgive me for any errors with formatting and my expressions. I am working on the following proof: Show $$x^n \geq ...
-1
votes
0answers
80 views

connection between $n$-th root of a number and n raised to the power of that number?

$$ \sqrt[n]{m} \, \,\, \,? \, \, \, \, n^m $$ n,m positive integers. I have noticed that $\sqrt{3}$ is the diagonal of the unit cube (two to the power three). I was wondering if this could have ...
4
votes
1answer
25 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 ...
3
votes
1answer
40 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 ...
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0answers
31 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 ...
0
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1answer
1k 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
80 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. ...
0
votes
1answer
89 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 ...
6
votes
2answers
268 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 ...
0
votes
0answers
27 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 ...
2
votes
6answers
57 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
1answer
31 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 $Pe=(Pt/N)(1-δ)^{\log_2 N}$ ...
0
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3answers
42 views
1
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1answer
49 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
1
vote
3answers
42 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?
0
votes
1answer
27 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
37 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?
2
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1answer
32 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
110 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
94 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
72 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
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1answer
26 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
311 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 ...
4
votes
1answer
23 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?
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vote
3answers
24 views