3
votes
1answer
44 views

matrix exponential limit

I'm having litlle trouble here to prove the following statement: "Let $A$ an $n\times n$ matrix (real or complex). Prove that $$\lim_{n \to \infty} \left(I + \frac{A}{n}\right)^{n} = e^{A}.$$ Now ...
0
votes
2answers
33 views

How to solve an exponential function with multiple addends

Our math teacher gave us the following exponential equation to solve: $3^x+10=2*7^x$ ...and I was stumped. Eventually, the solution given was to graph both sides and find their intersection using a ...
0
votes
0answers
54 views

Complex exponentiation

So I've got this question that is a bit difficult to ask, since it uses a term in my language that I can't properly translate into English. For $z\in\mathbb{C}^*$ and $a\in\mathbb{C}$ it would be ...
1
vote
2answers
24 views

Continuous compounding question

A population of rabbits starts out with $100$ rabbits. The growth rate is $11.7$% per day. Determine the exponential equation. Is it $$\mathbb {P(t)} = 100e^{11.7t}$$ Can you guys give me the ...
0
votes
5answers
76 views

How to solve $5^{n+2} - 5^{n-3} = -2500$ [closed]

How to solve $5^{n+2}- 5^{n-3} = -2500$
0
votes
3answers
60 views

Logarithm properties doubt

The problem is $\log (5.64)^4$. According to the properties and laws of exponents, $\log (m^r) = r \log (m)$. But since the exponent is outside of the parenthesis in this problem, does it solves by ...
1
vote
1answer
84 views

Find $x$ in $\large \,\, A^{A^{{{A^.}^.}^.}}= \,\,(\sqrt [x\cdot x\cdot x\cdot …] x)^{ (((1\cdot x+1)x +1)x +1)x+1…} $

If $\large \,\, A^{A^{{{A^.}^.}^.}}= \,\,(\sqrt [x\cdot x\cdot x\cdot ...] x)^{ (((1\cdot x+1)x +1)x +1)x+1...} $, and $\large \,\,A = (\sqrt[3]{3\sqrt 3 })^{\frac{\sqrt 3}{3}} $, find $x$ I have ...
0
votes
0answers
65 views

difficult inequality to prove

I need help proving this inequality is correct for a homework assignment: $$\displaystyle \left(\frac{13}{4}\right)^{n} \leq ...
2
votes
3answers
99 views

How can we differentiate $(x^{-1})^{({x^{-1})^{x^{-1}}}}$ wrt $x$?

How can we differentiate $(x^{-1})^{({x^{-1})^{x^{-1}}}}$ with respect to $x$?
2
votes
1answer
34 views

Problem finding limit - which function is asymptotically larger

I have a homework question, so please don't answer fully but I would appreciate a push in the right direction. Basically we need to figure out if $n^{n+\frac{1}{2}}e^{-n}$ is larger,smaller, or equal ...
1
vote
1answer
48 views

Simplify an expression.

Don't know how to do this. Simplify the expression, show steps: $$\large \dfrac {a^{-\frac 14}a^{\frac 32}}{a^{\frac 13}}$$ Write the answer using only positive exponents. Assume that all variables ...
1
vote
2answers
42 views

Proving that if $a>1$ and $x>y$ then $a^x>a^y$

I got this assignment for homework and I can't find this anywhere around the web. Prove that if $a>1$ and $x>y$ then $a^x>a^y$. I started the assignment but I'm not sure it's enough: ...
2
votes
2answers
280 views

How to compute $2^{\text{some huge power}}$

I have to compute $$2^{p-1} \mod p$$ and show by Fermat's little theorem that $p$ isn't prime. I know what the question is asking but I'm not sure how to reduce the exponent on $2^{p-1}$ to a more ...
0
votes
1answer
103 views

How to solve $5^n - 5^{n-3} = 5^{n-3} *124$

how is $$5^n - 5^{n-3} = 5^{n-3} *124$$ Can anybody provide a step by step solution.I will greatly appreciate if any online source for such material is provided. Regards
2
votes
1answer
56 views

What is the exponential for the matrix

What is the exponential for the matrix $$ \begin{pmatrix} 0 & -x & 0 \\ x & 0 & 0 \\ 0 & 0 & 0 \\ \end{pmatrix} $$ Is it $$ ...
2
votes
1answer
106 views

The domain of fractional exponents

Take the following: $$f(x) = x^{6/4}$$ The domain of this function is all real numbers. This function can be simplified to: $$f(x) = x^{3/2}$$ The domain of this function is all real numbers ...
1
vote
1answer
38 views

How is this possible? Can someone explain?

My teacher says that $W^{2/7}B^{5/7}=1$ is equivalent to $W^2B^5=1$. Can someone explain this rule to me? Am I always able to just take the variable and raise it to the numerator of the fractional ...
1
vote
1answer
291 views

Math induction problem (rules of exponents)

Hello I am doing some induction problems, I have to prove that $3^{k+1}-1$ is a multiple of 2. Suddenly they make this statement; $3^{k+1}$ is also $3 * 3^k$. Why is that?
1
vote
1answer
206 views

zero raised to infinity

I encountered a question where the only condition stated that $t>0$ and was then asked to compare these two quantities $0^t$ $t^0$ The scope of $t$ is $(0,\infty)$ and hence for infinity 1.) ...
2
votes
3answers
100 views

Why does $(-2^2)^3$ equal $-64$ and not $64$?

The title says it all. Why does $(-2^2)^3$ equal $-64$ and not $64$? This was on my algebra final, and I am completely stuck on how it works.
0
votes
1answer
74 views

k-fold matrix product

For $k \in \mathbb{N}$, $B,C \in \mathbb{R^{n,n}}$, given the matrices $B,C$ , calculate all powers $B^k$ and $C^k$ I'm a bit puzzled by this task. I assume it's supposed to practice handling ...
2
votes
1answer
100 views

Exponential equations involving natural numbers at power “x”

Find x : $$4^x+15^x=9^x+10^x(2^x-3^x)(2^x-3^x-5^x)$$
4
votes
1answer
61 views

Simplifying $y=2^{2/3} + 2^{-1/3}$

I am working on a calculus problem where I have to find the local minimum. The value I got was $$y=2^{2/3} + 2^{-1/3}.$$ I simplified it and got this: $$ y=2^{2/3} + \frac{1}{2^{1/3}}$$ ...
0
votes
1answer
41 views

Finding that probability of the event is small

Let $x_1, \ldots, x_n$ be Bernoulli random variables with the probability of success $P(x_i=1)=p$. Let $\epsilon>0$. Show that probability $$ P\left(\left|\sum_{i=1}^nx_i-p\right|> ...
3
votes
4answers
81 views

Proof of $\sqrt{2^{2^k}} = 2^{2^{k-1}}$?

It's quite easy to observe that for $k \ge 0$: $$ \begin{align} 2^{2^k} &= 4, 16, 256, 65536, \dots\\ \sqrt{2^{2^k}} &= 2, 4, 16, 256,\dots \end{align} $$ More in general: $$ \sqrt{2^{2^k}} ...
9
votes
2answers
288 views

Algebraic equation problem - finding $x$

$$(x^2 +100)^2 =(x^3 -100)^3$$ How to solve it?
1
vote
2answers
122 views

Simplifying negative exponents when there are multiple terms.

The problem is to simplify $$\left(\dfrac{1+3z}{3z}\right)$$ I know that when I have $$\dfrac{1}{x}$$ I can bring the $x$ from the denominator to the numerator by changing it to a negative power. How ...
8
votes
3answers
3k views

How do you compute negative numbers to fractional powers?

My teachers have gone over rules for dealing with fractional exponents. I was just wondering how someone would compute say: $$(-5)^{2/3}$$ I have tried a couple ways to simplify this and I am not sure ...
0
votes
2answers
83 views

Simplifying $ \;x({y^{3}}/{x^{4}})^{1/4}$

I’m a little unsure how to simplify the following expression: $$ x\left(\frac{y^{3}}{x^{4}}\right)^{1/4} $$ According to the answer, this should get you $\;\; x y^{3/4} x^{-1} = y^{3/4} $. My ...
0
votes
2answers
46 views

On exponentials of matrices

How can I prove that $|e^{A}|\leq e^{|A|}$? I guess i'm having trouble with the definition! A is a square (complex or real) matrix.
1
vote
2answers
199 views

$\det(\exp X)=e^{\mathrm{Tr}\, X}$ for 2 dimensional matrices

I want to prove that for $X\in M_2(\mathbb{R})$ the formula $\det(\exp X)=e^{\mathrm{Tr}\, X}$ holds, writing $X$ in normal form gives $X=PJP^{-1}$, where $J$ is the Jordan matrix, now $\exp ...
0
votes
3answers
270 views

Manipulating Exponents

I'm doing my homework and there are a couple of things that I am having trouble grasping. All my homework asks is that I simplify the exponents. For example: ...
1
vote
2answers
121 views

Simplifying $\frac{2^{n + 4} + 2^{n + 2} + 2^{n - 1}}{2^{n - 2} + 2^{n - 1}}$

I'm stuck in the follow equation: $$\dfrac{2^{n + 4} + 2^{n + 2} + 2^{n - 1}}{2^{n - 2} + 2^{n - 1}}$$ As all the bases are equal, I got $\dfrac{3n + 5}{2n - 3}$ Where I've to go now ? Thanks ...
0
votes
2answers
490 views

Graphing Fractional Exponents

$f(x)=x^\frac{5}{3}-5x^\frac{2}{3}$ is the same as : $f(x)=(\sqrt[3]x)^5-(\sqrt[3]{5x})^2$ Except, with the first equation, my calculator returns an error for negative values of $x$ (We are ...
0
votes
2answers
195 views

Simplifying exponents, multiplication, and addition

How can you get $10^{n+1}$ from $9\cdot 10^n+10^n$? This is part of a proof I am working on.
4
votes
3answers
149 views

Prove $\frac{(5^{x-1}+5^{x+1})^2}{25^{x-1}+25^{x+1}}=\frac{338}{313}$

Q. Prove $$\frac{(5^{x-1}+5^{x+1})^2}{25^{x-1}+25^{x+1}}=\frac{338}{313}$$ My try: expand and got: $$\frac{5^{2x-2}+2(5^{x^2-1})+5^{2x+2}}{5^{2x-2}+5^{2x+2}}$$ Now what? I find my pre-calculus ...
0
votes
1answer
62 views

Simplifying with negative exponents $(-11a^2)(-4a^{-7})$

$$(-11a^2)(-4a^{-7})$$ Can someone reformat, $a$ is second set of parenthesis is to the $-7$ power. Change to reciprocal so we get $$\left(\frac{1}{-4a}\right)^7 * \frac{11a}{1} = $$ confused ...
0
votes
1answer
68 views

Re-writing a logarithm to a power

Given: $$(4\ln x)^2$$ Is this simplified to $8\ln x$, (multiplying the expression by 2), $32\ln x$, (square $4$ ($16$), then $\ln x$ ($2\ln x$) and combine again), or something else? Just to be ...
3
votes
2answers
340 views

Non-integer exponents

Can you use noninteger powers Like is $x^{8.3} / x^{2.2} = x^{6.1}$?
1
vote
2answers
379 views

Missing exponent?

$\frac{256}{2^S}=64$ How would you solve for S?