2
votes
1answer
171 views

Solving Weird Exponential Equations

I am working on my math homework when I encountered a difficult problem. I simplified the equation and substituted smaller numbers to get this: $n*2^n>10$ I have tried standard algebraic ...
1
vote
1answer
26 views

Calculus / exponential

Find values of a and b so that $y = a·b^x$ and the line $y = x + 2$ are tangent at $x = 0$. I tried to substitute with the zero and it seem that the $B=1$ at all time but what about the $a$?
2
votes
3answers
155 views

Find positive integers $(x,n)$ such that $x^{n} + 2^{n} + 1$ is a divisor of $x^{n+1} +2^{n+1} +1$

Find all positive integers $(x,n)$ such that $x^{n} + 2^{n} + 1$ is a divisor of $x^{n+1} +2^{n+1} +1$ I encountered this question in one of my monthly assignments. Unfortunately, I don't know ...
6
votes
3answers
127 views

Primes as a difference of powers

Find the smallest prime that cannot be written as $$|3^a - 2^b|$$ EDIT: I forgot to mention that $a$ and $b$ are whole numbers. I tried to expand $3^a$ as $(2+1)^a$ using binomial theorem but ...
1
vote
1answer
43 views

Exponential problems

A ship embarked on a long voyage. At the start of the voyage, there were 300 ants in the cargo hold of the ship. One week into the voyage, there were 600 ants. Suppose the population of ants is an ...
2
votes
2answers
116 views

When is the power of a binomial equal to the sum of like powers of its terms?

Question: Under what circumstances/restrictions on $x$ and $y$ does $(x + y)^n = x^n + y^n$ given the value of $n$? That is, what can we tell about $x$ and $y$ from the value of $n$ and the equation ...
1
vote
5answers
109 views

solve the equation using logarithms (I think this is easy level)

Solve the equation for $x$ by using base 10 logarithms. $$16\cdot4^{2.5x}=9$$ EDIT: I made a typo (somehow... I was very far off!!) The correct equation is this: $$16\cdot4^{2.5x}=70$$ Can it be ...
0
votes
1answer
42 views

Homework help to rearrange formula

Given the equation $${V_m} = u(\ln {m_0} - \ln {m_8}) - g{t_f}$$ I need to solve for ${m_0}$ Here is what I have but it looks messy and I feel like there is sometihng wrong or a better way 1st ...
1
vote
2answers
39 views

Reciprocal of $7.5^{1-x}$

Ok my calculator tells me the reciprocal of $7.5^{1-x}$ is $0.1333\cdot7.5^x$. Can anyone explain the steps involved to get this manually? Is it along the line of the reciprocal of $7.5^1 + 7.5^{-x} ...
2
votes
0answers
21 views

Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $Dq(x) . Ax < 0$ for all $x \neq 0$

Show that $x' = Ax$ is an attractor if end only if there is a quadratic form $q$ positive definite such that $$Dq(x) . Ax < 0$$ for all $x \neq 0$ Definition: a linear system $x' = Ax$ called ...
8
votes
5answers
290 views

Why does $n^0 = 1$?

Why is it that $n^0 = 1$? I understand how $n^2 = n*n$ and how $n^1 = n$ but I can't understand why $n^0 = 1$.
3
votes
1answer
46 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
35 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
63 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
25 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
80 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
61 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
94 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
66 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
101 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
37 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
282 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
105 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
134 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
356 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
239 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
101 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
89 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
107 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
42 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
290 views

Algebraic equation problem - finding $x$

$$(x^2 +100)^2 =(x^3 -100)^3$$ How to solve it?
1
vote
2answers
134 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
4k 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
218 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
301 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
122 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
588 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
202 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
79 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
69 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
370 views

Non-integer exponents

Can you use noninteger powers Like is $x^{8.3} / x^{2.2} = x^{6.1}$?