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

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11
votes
9answers
2k views

Which of the numbers is larger: $7^{94}$ or $9^{91} $?

In this problem, I guess b is larger, but not know how to prove it without going to lengthy calculations. It is highly appreciated if anyone can give me a help. Which number is larger $$\...
0
votes
1answer
28 views

Interesting 4th order factoring question

$$ A = \frac{(4\cdot2^4 + 1)(4\cdot4^4 + 1)(4\cdot6^4 + 1)}{(4\cdot1^4 + 1)(4\cdot3^4 + 1)(4\cdot7^4 + 1)}$$ What is the value of $ \dfrac{113A}{61}$ ? So i tried factoring this $\dfrac{(4\cdot(x+1)^...
5
votes
7answers
364 views

Is $202^{303}$ greater or $303^{202}$?

Find without use of calculator which of the two numbers is greater $202^{303}$ or $303^{202}$. I think we have to do this with calculus because I got this question from my calculus book. I tried ...
-1
votes
1answer
26 views

Paramteric Curves and the exponents of $\cos$/$\sin$/$\tan$

Lets say we have the curve $\frac x7=\cos^7t$, $\frac y7=\sin^7t$ Now I know that $\sin^2x+\cos^2x=1$. So $\cos^2=(\frac x7)^{\text{some exponent}}$. What is that exponent? How do you work it out?
1
vote
3answers
723 views

How to prove the sum of combination is equal to $2^n - 1$

One of the algorithm I learnt involve these steps: $1$. define a set $S$ of $n$ elements $2$. form a subset $S'$ of $k$ choice from $n$ elements of the set $S$ ($k$ starts with $1$), which is ...
-2
votes
3answers
51 views

$ \sqrt[n]{b} =a \Leftrightarrow a^{n} =b$ [closed]

Why the two-way relationship is established: $$ b^{ \frac{1}{n} }=\sqrt[n]{b} =a \Leftrightarrow a^{n} =b$$
-3
votes
2answers
46 views

Solve Equation for n where n is the power ($3^n = \frac{1}{81}$) [closed]

I have the equation: $$3^n = \frac{1}{81}$$ And I need to find n. Can someone explain how I do this, with steps please (GCSE level)? TIA.
2
votes
3answers
75 views

Solve the system of equations $x+2^x=y+2^y$ and $x^2+xy+y^2=12$

$$x+2^x=y+2^y$$ $$x^2+xy+y^2=12$$ I'm having trouble solving this problem, please do not solve the entire problem, I just want a hint. I don't have any good idea.
0
votes
1answer
24 views

Log power rule problem

According to many parts of the Internet, this log rule is used. log(a^b) = b*log(a) The proof is: Now let's say I want to use the rule in a Cartesian ...
10
votes
5answers
125 views

Find the limit of $\frac{(n+1)^\sqrt{n+1}}{n^\sqrt{n}}$.

Find $$\lim_{n\to \infty}\frac{(n+1)^\sqrt{n+1}}{n^\sqrt{n}}$$ First I tried by taking $\ln y_n=\ln \frac{(n+1)^\sqrt{n+1}}{n^\sqrt{n}}=\sqrt{n+1}\ln(n+1)-\sqrt{n}\ln(n),$ which dose not seems to ...
5
votes
5answers
106 views

For integer $n>2$, $(n!)^2 > n^n$ [duplicate]

Problem: For integer $n>2$, show that $(n!)^2 > n^n$ My attempt: I tried using induction. For $n=3$, the given condition is satisfied. Let us suppose $k!^2>k^k$ for some $k\geq3$. Then, $(...
1
vote
1answer
13 views

Let $H = \{2^m : m \in \mathbb{Z}\}$ & define a relation $R$ on the set $\mathbb{Q^{+}}$ of positive rationals by $a\mathbin{R}b$ iff $a/b \in H$.

Let $H = \{2^m : m \in \mathbb{Z}\}$ and define a relation $R$ on the set $\mathbb{Q^{+}}$ of positive rational numbers by $a\mathbin{R}b$ if and only if $a/b \in H$. Prove that $R$ is an equivalence ...
3
votes
3answers
59 views

Prove that $24^{31}$ is congruent to $23^{32}$ mod 19.

According to my knowledge, to prove that $24^{31}$ is congruent to $23^{32}$ mod 19, we must show that both numbers are divisible by 19 i.e. their remainders must be equal with mod 19. Please correct ...
2
votes
2answers
38 views

Help solving the inequality $2^n \leq (n+1)!$, n is integer

I need help solving the following inequality I encountered in the middle of a proof in my calculus I textbook: $2^n \leq (n+1)!$ Where $\mathbf{n}$ in an integer. I've tried applying lg to both ...
1
vote
0answers
34 views

Unprovable identity over the integers

I was thinking about Tarski's problem, and was wondering what happens if we have a theory $T$ with two sorts $N,Z$ with intended interpretations $\def\nn{\mathbb{N}}$$\def\zz{\mathbb{Z}}$$\nn,\zz$ ...
1
vote
1answer
61 views

How to find $\frac{\mathrm{d}y}{\mathrm{d}x}$ when both number in front and exponent have fractions?

I'm not sure how to solve this: $\frac{5}{9}x^\frac{2}{3}$. I applied the product rule and have $\frac{2}{3}\frac{5}{9}x^{-\frac{1}{3}}$. $\frac{30}{9}x^{-\frac{1}{3}}$, then $\frac{9}{30}x^{\frac{1}...
10
votes
5answers
145 views

How do you solve $x^2 = \left(\frac 12\right)^x $?

I'm having trouble finding the steps to solve for $x$. The solutions to this equation are $x=-4$, $x=-2$, and $x=0.76666$ when solved graphically and through the solve function of a TI-nspire cx CAS. ...
0
votes
1answer
34 views

Fractional Exponents Confusion

Let a and b be natural numbers (not including zero). Is it true that will not equal for all possible solutions? For instance, if a=b the would always give an output of x (assuming you don't start ...
-4
votes
1answer
63 views

What is infinity to the zeroth power? [closed]

I am not happy with the answers posted to similar questions. For example, in: What is infinity to the power zero the accepted answer is 1, which is definitely wrong. I think the answer is any non-...
0
votes
0answers
12 views

Exponent operation over elements of $G$

I found a definition of an exponent operation over the element of $\mathbb{G}$ in this paper (page 4): $$ (g^a)^{\% b} = g ^{a \text{ mod } b}$$ I couldn't understand the rest of the paper (Decrypt ...
4
votes
1answer
151 views

Exponentiation of Gell-Mann Matrices

The exponentiation of Pauli vector $\vec \sigma=(\sigma_x,\sigma_y,\sigma_z)$ is trivial as we have the identity:$$e^{ia(\vec n\cdot \vec \sigma)}=I cos(a)+i(\vec n \cdot \vec \sigma)sin(a)$$ I have ...
1
vote
1answer
30 views

Subtraction of large numbers with exponents

Is there an easy way to break down the below formula? And make it easy to calculate it mentally without the use of a calculator? $108^2 - 92^2$ I know this is probably very basic, but I cant ...
5
votes
1answer
80 views

Combinatorics problem that deals with trigonometric functions

If $m$ and $p$ are positive integers and $m \geq p$, then show that $${m \choose 0}+{m \choose p}+{m \choose 2p}+{m \choose 3p}+\cdots$$ has value $${2^m \over p}\left(1+\sum_{k=1}^{\left \lfloor {...
1
vote
1answer
20 views

negative fraction exponent and division

Quick question on how to handle negative fraction exponents when differentiating: I have this problem to differentiate. $$x^{2/3} + y^{2/3} = 1$$ So my textbook and I both did the first thing the ...
0
votes
1answer
47 views

How to calculate a definite integral with complex numbers involved?

I'm trying to calculate this integral, and I find it difficult when coping with complex numbers. $$ f(k) = \int_{lnK}^{\infty} e^{ikx} (e^{x}-K) dx =(\frac{e^{(ik+1)x}}{ik+1}-K\frac{e^{ikx}}{ik})|_{x=...
0
votes
1answer
35 views

Explanation of i to the i power? [duplicate]

Could somebody give me a good explanation for how $i^i$ works? I'm a junior and just now getting to this. I'm also too hard pressed for time to dive into exploring it myself.
1
vote
1answer
22 views

$2^i \equiv 2^j \pmod n$ implies $2^{j−i }\equiv1$ if $n$ is odd; also if $n$ is even?

Show that, if $0 \leq i < j$ and $2^i \equiv 2^j \pmod n$ and $n$ is odd then $2^{j−i} \equiv 1 \pmod n$. Is this necessarily true if $n$ is even? I have tried to prove this by using Fermat's ...
2
votes
1answer
18 views

How to find the highest [natural] radix base of a given number with a natural output

Like the title says, I'm trying to make a program that finds the highest natural radix of a given number with a natural output. My program works, but it loops every number possible number up to a ...
4
votes
0answers
180 views

What is the middle digit of $9^{99}$?

Find the total number of digits and the digit in the middle of $9^{99}$, $\it{without}$ actually calculating any other digit of the number. PS: according to defuse.ca/big-number-calculator.htm, the ...
2
votes
3answers
158 views

$(-1)^{\sqrt{2}} = ? $

This popped up when I was thinking about $$(-1)^{\frac {p}{q}}$$ where $ p $ and $q$ are integers such that $\gcd (p,q) = 1$ If $p$ is even : $(-1)^{\frac {p}{q}} = +1$ If $q$ is even : $(-1)...
0
votes
1answer
36 views

Powers with complex/negative bases

If x can be a positive real number (for example a fraction with a numerator and denominator), then why does the following relationship hold true only if and only if a and b are strictly positive real ...
1
vote
1answer
35 views

Limit with logarithm: $\lim_{n \to \infty} \frac{n^\alpha}{\ln^\beta n}$

What is the limit $\lim_{n \to \infty} \frac{n^\alpha}{\ln^\beta n }$ (ln=natural logarithm) for alfa real and less than zero? I found out it is zero for $\beta\ge0$, since then you can use the ...
1
vote
1answer
42 views

The exponent of $11$ in the prime factorization of $ 300!$ is___.

The exponent of $11$ in the prime factorization of $ 300!$ is $27$ $28$ $29$ $30$ My attempt: According to Exponent of $p$ in the prime factorization of $n!$ $\left\lfloor\dfrac{300}{11}\...
1
vote
1answer
24 views

Digit-sum division check in base-$n$

Several years ago now I realised that for any natural numbers $x$ and $y$ you could write $$x^y=(x-1) \left(\sum_{i=0}^{y-1}x^i\right)+1$$ This shows that $x^y-1$ will always be divisible by $x-1$, ...
0
votes
0answers
19 views

Exponent of x in the prime factorization of y?

This is a simple one. I recently came across the following phrase: "$(x)_y =$ the exponent of $p_y$ in the prime factorization of $x$, for some $x, y<0$" I know what prime factorization is, and ...
-5
votes
1answer
59 views

Proof that $0^0 \neq 1$ [closed]

Suppose that $t = \sqrt{t}^{\sqrt{t}}$, then, it follows that; $$ t^{\sqrt{t}} = \sqrt{t}^{t} \\ \frac{1}{2}t\ln{\left(t\right)} = \sqrt{t}\ln{\left(t\right)} \\ \ln{\left(t\right)}\left[\frac{1}{2}t ...
2
votes
1answer
682 views

Why does $\frac{1}{4}x^{-3/4} = \frac{1}{4x^{3/4}} = \frac{1}{4\sqrt[4]{x^3}}$?

This is taken from Khan Academy, I don't understand how these equate: $$\frac{1}{4}x^{-3/4} = \frac{1}{4x^{3/4}} = \frac{1}{4\sqrt[4]{x^3}}$$ How come the minus was remove from the original exponent?...
0
votes
1answer
10 views

Name for inequality about sums of exponents with same base

Is there a name for the following inequality regarding sums of exponents which share a base? $$\text{For all integers $b \geq 2$, $n \geq 1$,} \\ \sum_{i=0}^{n-1}{b^i} < b^n$$
0
votes
0answers
16 views

Is it possible to transform different interest rates (applied over different times) into one interest rate?

I have a fixed income asset that has the following payment schedule. Is it possible to transform the multiple interest rates being applied to different periods into one interest rate and one period? I ...
3
votes
2answers
57 views

Enough differences of powers of natural numbers equals a constant

Let $n_k$ be a sequence. We create a new sequence by taking the difference of consecutive terms in $n_k$. So the terms of the new sequence $a_k$ is defined as $a_i=n_{i+1}-n_i$. This is the difference ...
2
votes
1answer
64 views

Scalar by matrix derivative $\frac{d {\rm tr} (e^{\bf X} {\bf A})}{d {\bf X}}$

I'm trying to figure out what is the scalar by matrix derivative $\frac{d {\rm tr} (e^{\bf X} {\bf A})}{d {\bf X}}$ equals to? I know that $\frac{d {\rm tr} (e^{\bf{X}})}{d \bf{X}} = e^\bf{X}$ and $\...
0
votes
3answers
36 views

Exponentiation of Diagonalizable Matrix

Wikipedia says that "If $A = UDU^{−1}$ and D is diagonal, then $e^{A} = Ue^{D}U^{−1}$" Why is this the case? I understand that $e^D$ yields a matrix where $M_{i,j} = e^{D_{i,j}}$, but how is it ...
2
votes
4answers
52 views

Simplifying and taking the limit

I would like to compute the following limit $$\lim_{n\to\infty}\frac{10^{n^{2}}}{10^{(n+1)^{2}}}$$ but I'm having a hard time simplifying. Can anyone explain to me the properties of these exponents? ...
0
votes
0answers
24 views

Given complex numbers raised to real powers, what do imaginary part = 0 level sets look like?

Suppose you lay flat (horizontally) the Argand plane, with x real. Consider the vectors v in that plane. Then let z, the vertical axis, be the real part of the complex solution to v^t, with t real. So ...
0
votes
1answer
37 views

How to solve an equation with the unknown variable both in the exponent and in linear summand?

How to solve the following equation? $$e^{dz} - mdz = 1$$ where $z$ is the unknown variable, the others are constants, $\exp(x)$ is taking $e$ to the power $x$ . I am interested in real solutions ...
0
votes
1answer
28 views

Simplifying Large Bases with large Exponents

I'm told to find: $105 308^{7125} \pmod {11}$ I'm not exactly sure how to go about calculating this. I know that I could split the exponent into multiples of it, for instance. $7125 = 7 * 10 * 10 * ...
0
votes
3answers
45 views

How to differentiate x^(1/x)?

How to differentiate the following? $$x^{\frac{1}{x}}$$ (I know the answer is $\frac{1-\ln(x)}{x^{2-\frac{1}{x}}}$, but I do not understand how to get there) Attempt at solution I believe the ...
0
votes
2answers
32 views

Order of exponents matters?

I always thought that $(a^b)^c=a^{bc}$... I am confused about why the order of exponents seems to matter in this particular case: $((x-0.5)^2)^{1/3}$ is not the same as $(x-0.5)^{2/3}$ ...
0
votes
1answer
31 views

approximation of binomial coefficient by exponentiation

Show that inequality holds for every n, k. $$\ { n \choose k} \leq \frac{n^k}{k!}. ( \, 1-\frac {k}{2n} ) \,^{k-1} $$ I used Stirling's formula but I stucked, which is $$\ { n \choose k} = \frac{n^...