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Questions tagged [exponentiation]

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

2
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1answer
44 views

Subtracting exponents properties?

Background : I was reviewing some practice problems for a small local math competition, and I don't understand how the give solution works. I don't know how this, $$9^{x+2} - 9^x=240$$ is the same ...
0
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2answers
170 views

Prove or disprove that there is a rational number $x$ and an irrational number $y$ such that $x^y$ is irrational.

In solving the following problem: Prove or disprove that there is a rational number $x$ and an irrational number $y$ such that $x^y$ is irrational. I let $x=2$ and $y = \sqrt 2$, so that $x^y = 2^\...
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1answer
29 views

How was 'DF' calculated? [closed]

Asking formulae to how to calculate variable DF ..I would like to know how 'DF' was calculated as below my screenshot: enter image description here unknow: DF, DFCF Known: # of Year, Rate=10%, FCF, ...
0
votes
1answer
51 views

Show whether or not the functions $f(x,y)=|x|^y$, $g(x,y)=|x|^{|1/y|}$ have limits at $(0,0)$.

Show whether or not the functions $f(x,y)=|x|^y$, $g(x,y)=|x|^{|1/y|}$ have limits at $(0,0)$. By the answers only the latter has a limit at $(0,0)$ but I don't know how to prove.
1
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1answer
37 views

Connection between powers of $i$ and derivatives of $\sin$ and $\cos$?

There are plenty of connections between trigonometry and the complex plane. One particular comparison seems striking to me, and that’s the four-step cycle inherent in exponentiating $i$ to various ...
0
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1answer
23 views

How to write $12 (N / m)^2$ in long form

I want to write the value $12 (N/m)^2$ in long form like "12 Newtons per meter squared". However, I believe that because of the order of operations, my long form value will be interpreted as $12 N/(m^...
2
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1answer
60 views

$(-1)^3$ has different results when evaluated as $(-1)\times(-1)\times(-1) = -1$ vs $((-1)^2)^{3/2} = 1$. Which is correct?

I know that $$(-1)^3=(-1)\times(-1)\times(-1)=-1 \tag{1}$$ but also $$(-1)^3=((-1)^2)^{3/2}=1^{3/2}=1 \tag{2}$$ So which gives the correct value of $(-1)^3$?
1
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1answer
73 views

A problem in algebra: how does $-1=1$? [duplicate]

I have algebra problem from a friend, that is 1=-1!!! because $$-1=-1^{3}=-1^{^{\frac{6}{2}}}=\sqrt{(-1)^6}=\sqrt{1}=1$$ I can not see what is wrong with this? I will appreciate it any help.
0
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2answers
46 views

What is the base of this exponent?

Given the following, what is the value of $ 2b^5 $? $$ b = 5 $$ $$ 2b^2 $$ I'm confused as to whether the exponent applies to $ 2b $ or just $ b $. Thus, does $ 2b^2 $ equal 50 or 100? What is the ...
2
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3answers
53 views

General formula of $\lim_{n \to \infty} (1+a_n)^{b_n}$ where $b_n \to \infty$

What is the general formula of $$\lim_{n \to \infty} (1+a_n)^{b_n} \quad \text{where} \quad \lim_{n \to \infty} b_n = \infty?$$ For example we have $$\lim_{n\to \infty} \left(1 + \frac1n\right)^n = e$...
1
vote
1answer
40 views

Is there a shorter way to say “b raised to the n-th power”

$b^n$ Alternatives that I know are correct: "b raised to the power of n" "the n-th power of b" Can I just say "b raised to n" or "b to n" or are these technically wrong?
0
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1answer
51 views

Absolute value of a complex expression

I've been stuck on this problem for a while: $$\vert 1-e^{-i2\pi f}+.5e^{-i2\pi f\cdot 2}\vert^2,$$ where $i =$ the imaginary unit, $(2\pi f) =$ a real value, and $(2\pi f\cdot 2)=$ a real value. I ...
0
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2answers
84 views

How to express 80^2/3 as a product of powers of prime numbers

So the given was: $(80)^\frac23(25)^\frac32$ and I was told to simplify and express as a product of powers of prime numbers. Now I'm not very familiar with this product of powers of prime numbers so ...
1
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3answers
81 views

Calculating $\lim\limits_{j \to \infty} \frac{j^{j/2}}{j!}$

\begin{align*} & \lim\limits_{j \to \infty}{j^{\,j/2} \over j!} \\ \end{align*} This problem is from a real analysis textbook in the chapter on the natural log and properties of exponents. I'm ...
2
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1answer
58 views

Finding the inverse of my f(x) function that includes a fractional exponent (with a variable in the fraction)

Im a noob mathematician. I am trying to find the inverse of this function: $$F(x) = \frac{x + 299 \cdot 2^\frac{x-1}{7}}{4} $$ In the same way that: $$f(y) = ½(y - 1)$$ is the inverse of: $$f(x) = ...
5
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2answers
94 views

How do I solve for $x$ in $ab^x-cd^x=e$?

For example: I know that $5*1.2^{13}-8*1.1^{13}\approx26$. How do I find the exponential value (13 in the previous example) that would equate to 26 exactly? The answer should be something close to 13....
0
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3answers
30 views

solve variable in base

I am asking very petty question. I am confused to solve following equation. Answer should be 9.03. When I calculate, I constantly get different answer (696.4). How would you solve? then I wanna know ...
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5answers
67 views

Why is $a^{-x}$ defined to be equal to $\frac{1}{a^x}$? [duplicate]

I have searched the reason behind this definition in two textbooks and haven't found any. They just state that this is the definition but don't ever give any motivation for why this is truth. Edit: ...
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2answers
79 views

If $2^a=3^b$ find $\frac{a}{b}$ [closed]

I tried many different things but still couldn't solve it. Could you please give me a clue?
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6answers
79 views

Why is $ a^x = e^{x \log a} $?

Why is $ a^x = e^{x \log a}$, where $ a $ is a constant? From my research, I understand that the the natural log of a number is the constant you get when you differentiate the function of that ...
6
votes
7answers
169 views

Proof that $x^{(n+4)} \mod 10 = x^n \mod 10$

While solving a programming challenge in which one should efficiently compute the last digit of $a^b$, I noticed that apparently the following holds (for $n > 0$) $x^{(n+4)} \mod 10 = x^n \mod 10$ ...
4
votes
4answers
216 views

How to solve equations of this type [closed]

My math teacher gave us some questions to practice for the midterm exam tomorrow, and I noticed some of them have the same wired pattern that I don't know if it has a name: Q1. If: $x - \frac{1}{x} = ...
1
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1answer
34 views

Sum of a series of a number raised to incrementing powers— for any number

Sum of a series of a number raised to incrementing powers. I have a sub question on this. As I dont have reputation to ask for in comment. The formula being derived for series of summation of powers ...
1
vote
1answer
88 views

How can I convert $(a + bi)^{c+di}$ to polar form?

Specifically, I need to convert $(-8i)^{2+πi/3}$ to polar form. I understand I need to use Euler's formula, $e^{\theta i} =\cos \theta + i \sin \theta$, but I'm not sure about the full process. ...
0
votes
0answers
42 views

determining x and y offset of a curve segment

Good afternoon. I have a curve with the formula: $$f(x)=x^{2.2}$$ I have curve segments which are offset from the above formula that have the general formula ($a$ represents horizontal displacement ...
5
votes
2answers
1k views

If $3^x = 5$, $5^y = 10$, $10^z = 16$, then what is $3^{xyz}$?

Can't post images so I'll type it here: $$3^x = 5,\qquad 5^y = 10,\qquad 10^z = 16$$ Then what is $3^{xyz}$? I've spent like an hour trying to solve it and I failed. Help would be super duper ...
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4answers
67 views

$(\sqrt{3} + i)^{50}$ in exponential and cartesian form

Having trouble understanding the solution to the following question: Put $(\sqrt{3} + i)^{50}$ in exponential and cartesian form. I know the answer cartesian form is: $$\frac{2^{50}}{2} + \frac{2^{...
2
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4answers
120 views

If the number $“1”$ is written at the beginning, at least how many steps should be taken to reach $2^{2018}?$

I have a problem understand this math problem: Write a number on the board. This number is either multiplied by $2$ or raised to a square. If the number $"1"$ is written at the beginning, at least ...
0
votes
1answer
23 views

Proving that $\lfloor{i/2^h}\rfloor = \lfloor \lfloor\cdots \lfloor i/2\rfloor/2\cdots\rfloor/2\rfloor$

I am trying to prove that $$\left\lfloor{\frac{i}{2^h}}\right\rfloor$$ equals to performing a series of $h$ operations of $$\left\lfloor\frac{\left\lfloor\frac{\left\lfloor\frac{i}{2}\right\rfloor}{2}\...
2
votes
1answer
196 views

If $\frac {a}{3^{x-1}}=\frac{b}{3^{y+2}}=\frac{c}{3^{z-1}}=\frac 15\;$ then which of the following equals $a×b×c$?

The problem is: If $\frac {a}{3^{x-1}}=\frac{b}{3^{y+2}}=\frac{c}{3^{z-1}}=\frac 15,\;$ then which of the following equals $a×b×c$ ? A) $\frac {1}{375}$ B) $\frac{1}{125}$ C) $\...
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votes
2answers
40 views

($x=\sqrt[n] m ≠ x=m^\frac 1n $) when m is an even number

Note: In this question I am using the radical symbol to denote all roots of a number, not just the positive one. $$x=\sqrt[n] m ≠ x=m^\frac 1n $$ when m is an even number for example:- $$x=\sqrt4$$ $...
0
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1answer
15 views

Finding the minimum number to be multiplied to reach the closest kth root

I was trying to solve this coding question. You are given an integer $N$ and a value $k$. You need to find a minimum number $X$ which when multiplied to $N$ results in the value of $N^{1/k}$ as an ...
0
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1answer
43 views

Induction on powers of powers.

So I would like to prove that $ a_n \; | \; a_{n+1} - 2 $, where $a_n = 6^{2^n} + 1 $ Now I know I need to do this by induction and so I begin by showing this is true for the base case. (Proposition)...
1
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3answers
34 views

${x^4}$ as “tesseracting” a number $x$ [closed]

So, this strange thought popped up into my head. You know how we call ${x^2}$ squaring due to the fact that what you're essentially doing is finding the area of a square with side length $x$? The same ...
3
votes
2answers
82 views

Tetration of non-integers: is there something wrong with this approach?

I'm trying to figure out a formula for tetration that will work for non-integer heights. I know the usual recurrence relation for tetration ($x \in \mathbb{R}, \text{ }n \in \mathbb{N})$: $${^{n}x} =...
1
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2answers
39 views

Trying to prove an equation

I would like to receive some help about the next problem. The problem: I'm trying to prove the next equation: $$\sum_{k = 0}^{n} \frac{(-1)^{-k}}{k!(n - k)!} = 0 \quad, n = 1, 2, ...$$ My work ...
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2answers
27 views

Smallest integer power for an inequality to hold

so I have this inequality: Given integers $m, k\geq1$. $$2^{m/k} > \frac{3}{2}$$ I'm interested in finding the smallest integer power $m$, as a function of $k$, that will make this inequality ...
2
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4answers
73 views

Comparing $\ln 1000$, $\sqrt[5]{1000}$, $3^{1000}$, and $1000^{15}$ without calculator

In my Pre-Calculus class we were given the following problem: Put the following four values in order from smallest to largest: $\ln 1000$, principal $5$th root of $1000$, $3^{1000}$, and $1000^{15}$...
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1answer
24 views

Proof of exponentiation law

I want to prove this : $(ab)^n = a^nb^n$ with a, b and n real numbers. I know how to prove this when n is an integer but not when n is a real number. I really don't know where to start to prove this. ...
8
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5answers
636 views

Proving that $\lim_{n\to\infty} \left(1+\frac{1}{f(n)}\right)^{g(n)} = 1$

I want to prove that $$\lim_{n\to\infty} \left(1+\frac{1}{f(n)}\right)^{g(n)} = 1$$ if $f(n)$ grows faster than $g(n)$ for $n\to\infty$ and $\lim_{n\to\infty} f(n) = +\infty = \lim_{n\to\infty}g(n)$. ...
0
votes
1answer
24 views

Whether a Pathological Set Might Exist that could Foil a Theorem about the Exponential Function

There has been some discussion recently about the theorem $$\lim_{n \rightarrow \infty} \prod_{i = 1}^n \left( 1 + \frac{x_i}{n} \right) = \exp \left( \lim_{n \rightarrow \infty} \frac{1}{n} \sum_{i =...
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0answers
30 views

Can raising a number to an irrational power have infinite solutions?

$a^{\frac{1}{2}}$ is generally considered to be the positive square root of $a$, but it also makes sense (depending on context) to consider it to be multivalued, returning all square roots of $a$ ...
2
votes
2answers
70 views

Which of the following has the greatest value

Which of the following has the greatest value? a) $2^{64}$ b) $4^{63}$ c) $8^{34}$ d) $16^{17}$ I tried finding a pattern among exponents and their is none. but there is a pattern in base, but I'...
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vote
1answer
30 views

What to do if a number becomes bigger than 10 on scientific notation when performing arithmetics with it?

For example, I was adding (9.99 x 104)+ (9.99 x 105). I raised the first number's exponent from 4 to 5 by dividing it by 10 once more, thus getting 0.999 and the 105 exponent as a way to reverse it (...
5
votes
6answers
111 views

The expression: $-5^2$

A few people I met were debating over if $-5^2 = 25$ or $-25$. From my experience, we assume operator precedence and get $-25.$ People are telling me however, calculators are flawed, the real answer ...
0
votes
3answers
47 views

Basic Mathematics, Rules for Multiplication trouble

Doing some self study from the text Basic Mathematics by Serge Lang I ran into an exercise question which I can't seem to wrap my head around. The question is: Express the following expressions in ...
0
votes
1answer
30 views

A question about exponential functions: $a^b>b^a$ for $b>a>e$ [closed]

How can we prove that for $b>a>e$ ($e$ being the Euler’s number), $a^b$ is greater than $b^a$?
3
votes
1answer
31 views

Representing positive integers as floor of integer powers of real number.

Does there exist real $a$ such that for every positive integer $c$ there exists integer $b$ such that $\lfloor{a^b}\rfloor = c$?
0
votes
2answers
46 views

Exponent laws for rational numbers

this seems to be easy but I dont manage to prove it properly. You are only allowed to use the following laws: $$\left(z^m\right)^n=z^{m\cdot n}\quad \text{for $n,m\in \mathbb N$}$$ and $$z=a^{\frac ...
0
votes
2answers
34 views

Larger value with right associative tetration?

Given right associative tetration where: $^{m}n =$ n^(n^(n^…)) And a situation such as: $^{m}n = y$ $^{q}p = z$ What is a practical way to calculate which of $y$ and $z$ are larger? I'm ...