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

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0
votes
1answer
31 views

$4^{3a-1}-5^{2b-3}=0$ find a in terms of $b$

If $4^{3a-1}-5^{2b-3}=0$ then find a using $b$. My Attempt:we know that $2^{6a-2}=5^{2b-3}$ with this way we can find a value for $a$ and $b$ if both sides are zero if we can find another value for ...
0
votes
2answers
16 views

Image: proof re fractional exponents

Can someone help me prove that $(a^m)^{1/n} = (a^{1/n})^m$ per the textbook excerpt captured in the image?
9
votes
8answers
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 ...
2
votes
3answers
3k views

Trace of the matrix power

Say I have matrix $A = \begin{bmatrix} a & 0 & -c\\ 0 & b & 0\\ -c & 0 & a \end{bmatrix}$. What is matrix trace tr(A^200) Thanks much!
0
votes
2answers
33 views

log to exponential form, but with number in front of log

So I understand how to put a log equation into exponential form. For example, $y = \log_2(x)$ is $2^y = x.$ However, I don't understand what to do when there is a number in front of $\log$, such as ...
1
vote
2answers
71 views

$8^a=3$ and $3^b=5$ and $10^c=5$ then find $c$ in terms of $a$ and $b$.

if $8^a=3$ and $3^b=5$ and $10^c=5$ then find $c$ using $a$ and $b$. My Attempt: if $8^a=3$ and $3^b=5$ then we can say that $8^{ab}=5$ and then we have $2^{3ab}=10^c$ but i cant solve this ...
3
votes
0answers
79 views

Solving an equation with infinite exponents [closed]

How would I find the exact value of this infinite power tower? $$(1+2^{0})^{(1+2^{-1})^{(1+2^{-2})^{.^{.^{.}}}}}$$ I have found a decimal expansion of the number through calculation of the first few ...
2
votes
1answer
37 views

Exponential of a symmetric matrix

Let $A$ be a real, symmetric and positive definite matrix and suppose $B$ is a real symmetric matrix such that $\exp(B) = A$. Is $B$ unique? The solution of my homework sheet says that $B$ is ...
75
votes
15answers
5k views

math fallacy problem: $-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1$?

I know there is something wrong with this but I don't know where. It's some kind of a math fallacy and it is driving me crazy. Here it is: $$-1= (-1)^3 = (-1)^{6/2} = \sqrt{(-1)^6}= 1?$$
1
vote
1answer
28 views

simple question, need help

Can someone tell me where does 1 come from on the end, this got me really confused.
0
votes
1answer
41 views

How the derivatives are different if sign changes.

I have this expression $$\frac{1}{(1 - x) ^ 2}$$ I need the derivative of this expression. So I calculated it, no big deal. However something has crossed my mind. Mathematically $(1 - x) ^ 2 = (x - 1) ...
0
votes
1answer
16 views

Exponential equation with square variable as an exponent?

I am trying to solve the following exponential equation where the variable is squared. Most likely it is not difficult, but I am just missing the technique: what is the way to solve an exponential ...
2
votes
1answer
40 views

Is there a simple solution to these 2 equations without trying all possible values?

Now I have two equations and the computation is in a finite field GF(p), where p is a prime. $x, y$ are unknown, and $a, b$ are known. ($0<y<p-1$, and $0<ax<p-1.$) $\begin{cases} x^y ...
0
votes
0answers
40 views

If $\log_510=\log_7x(\log_nm)$ then the values of x,m and n are?

I have the question that if $\log_510=\log_7x(\log_nm)$ then values of $x$,$m$ and $n$ are? This question looks easy but i tried to get the expression down to the form $$\log_ab=\log_ac\tag{1.}$$ and ...
-3
votes
1answer
27 views

$\forall x,y\in \mathbb{R}\colon\forall n\in \mathbb{N}\colon [Odd(n)\lor Even(n) \land y\geq 0\implies [x^\frac{1}{n} =y\iff x=y^n ]]$ [closed]

Prove the following theorem : $\forall x,y\in \mathbb{R}\colon\forall n\in \mathbb{N}\colon [Odd(n)\lor Even(n) \land y\geq 0\implies [x^\frac{1}{n} =y\iff x=y^n ]]$ Thank you :)
-1
votes
2answers
65 views

Simplify: $S=3^{1/3}\cdot 7^{1/4}$

Simplify: $$S=3^{1/3}\cdot7^{1/4}$$ How is it possible to simplify this? The exponents are completely different.
0
votes
2answers
34 views

What is the units digit of the product of several numbers

This problem is quite challenging to me. I know the answer should be an even number, but not know how to solve the problem. Thank you for help! What is the units digit of the product $2^1·2^2· 2^3 ...
1
vote
3answers
54 views

How many zeros are at the end?

My answer to the following problem is 1337 (from the first one) +2 (from the second one) + 4646 (from the last one) = 5985. But it is different from what in answer sheet. I wonder whether I get the ...
18
votes
4answers
6k views

Finding $\int x^xdx$

I'm trying to find $\int x^x \, dx$, but the only thing I know how to do is this: Let $u=x^x$. $$\begin{align} \int x^x \, dx&=\int u \, du\\[6pt] &=\frac{u^2}{2}\\[6pt] ...
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 ...
3
votes
3answers
154 views

Which one is greater $\left(5/2\right)^{2/5}$ or $\left(7/2\right)^{2/7}$?

I have a question here: Which of $\left(5/2\right)^{2/5}$ and $\left(7/2\right)^{2/7}$ is greater? I tried comparison by the function $y=x^{1/x}$ and found the derivative as follows ...
6
votes
5answers
237 views

Given $a>b>2$ both positive integers, which of $a^b$ and $b^a$ is larger?

Given $a>b>2$ both positive integers, which of $a^b$ and $b^a$ is larger? I tried an induction approach. First I showed that if $b=3$ then any $a \geq4$ satisfied $a^b<b^a$. Then using that ...
5
votes
7answers
342 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?
-3
votes
2answers
87 views

Why exp(x) is so special that- [closed]

$\exp(x)=\int \exp(x) \; \text{d}x$ = derivative of $\exp (x)$ with respect to $x$. I'm curious to know this? Do other such functions exist?
1
vote
3answers
703 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 ...
4
votes
1answer
130 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 ...
-2
votes
3answers
51 views

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

Why the two-way relationship is established: $$ b^{ \frac{1}{n} }=\sqrt[n]{b} =a \Leftrightarrow a^{n} =b$$
9
votes
5answers
120 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 ...
-3
votes
2answers
45 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.
21
votes
5answers
4k views

$\sin(A)$, where $A$ is a matrix

If $A$ is an $n\times n$ matrix with elements $a_{ij}$ $i=$i'th row, $j=$j'th column. Then $e^A$ is also a matrix as can be seen by expanding it in a power series.Is $e^A$ always convergent and ...
2
votes
1answer
1k views

How to deal with negative exponents in modular arithmetic?

So I think I understand how to calculate something like $(208\cdot 2^{-1})\mod 421$ using extended euclidean algorithm. But how would you calculate something like $(208\cdot2^{-21})\mod 421$? ...
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 ...
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
57 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 ...
37
votes
4answers
12k views

Can you raise a number to an irrational exponent?

The way that I was taught it in 8th grade algebra, a number raised to a fractional exponent, i.e. $a^\frac x y$ is equivalent to the denominatorth root of the number raised to the numerator, i.e. ...
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
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 ...
1
vote
0answers
33 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$ ...
10
votes
5answers
142 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 ...
5
votes
2answers
470 views

What exactly are those “two irrational numbers” $x$ and $y$ such that $x^y$ is rational? [duplicate]

It's possible to prove nonconstructively that there exists irrational numbers $x$ and $y$ such that $x^y$ is rational, but that proof only proves that such numbers exist and does not specify what they ...
1
vote
3answers
64 views

Easy exponentiation method

Is there a simple way of solving, say, $x^{3/2}$? For example, one way of solving $16^{3/2}$ is to calculate the square root of $16^3$, but I was wondering if there is a simpler mental trick for ...
-4
votes
1answer
58 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 ...
45
votes
11answers
8k 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 ...
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 ...
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 ...
1
vote
1answer
19 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 ...