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

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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
60 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 ...
0
votes
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
votes
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
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 ...
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 ...
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 ...
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 ...
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 ...
1
vote
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 ...
-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 ...
43
votes
11answers
6k 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
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 ...
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 ...
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
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 ...
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
votes
3answers
42 views

Why does $e^{i5 \pi /6}$= -$e^{-i \pi /6}$?? [closed]

Why is $e^{i5 \pi /6}$ the same as -$e^{-i \pi /6}$?? Thanks.
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 ...
1
vote
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
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?
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?
2
votes
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
109 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
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 ...
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
vote
3answers
24 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
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?
2
votes
3answers
52 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
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 ...
3
votes
1answer
41 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
50 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
37 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
51 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
544 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.