2
votes
2answers
45 views

Find the value of $\frac{S_{5}S_{2}}{S_{7}}$

If $a$, $b$, $c$ $\in \mathbb R$, we define $S_{k}=\frac{a^k+b^k+c^k}{k}$ (where $k$ is a non-negative integer). Given that $S_{1}=0$, find the value of $$\frac{S_{5}S_{2}}{S_{7}}$$ I tried: ...
2
votes
3answers
88 views

Solve exponential equation $3^x= 2^x+2$

How do we solve this? I can't think of an easy way.. Is there any way to solve it without using newton's method or other approximations? $3^x=2^x+2$
2
votes
1answer
175 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 ...
0
votes
1answer
11 views

Finding an exponential formula passed upon the start and end points.

I'd like to create pricing curve that's based upon a reverse exponential function. I know the starting point and ending point, but don't know how to create the curve in between. For example, say for ...
5
votes
1answer
146 views

Solving the equation $a ^ b + b ^ a = 200$

Find $a$ and $b$, $a ^ b + b ^ a = 200$ One of the answers is $a = 1$ and $b = 199$. Lets say $a, b$ belongs to $\mathbb{R}$ then there will be many solutions, for each $a$ there exist $b$, in ...
0
votes
1answer
44 views

Methods for solving equations with exponents?

In the following equation, capital letters represent arbitrary real numbers that are constant with respect to $x$: $$A\left(x+B\right)\left(1 + \frac{C}{x+D}\right)^E + Fx + G = 0$$ I'm trying to ...
1
vote
3answers
66 views

How to prove that $\frac{a^n}{a^m}$ is equal to $a^{n-m}$? [closed]

How to prove that $\dfrac{a^n}{a^m}$ is equal to $a^{n-m}$? Thank you in advance.
2
votes
1answer
24 views

How to simplify this expression that contains exponential terms?

In a multiple choice exam , I encountered the following question. The answer to the question is $$ \frac{17}{8}.$$ The question is: $$\frac{16^{x+1}+4^{2x}}{2^{x-3}8^{x+2}} \text{ is ? }$$
0
votes
1answer
25 views

Economics question. Rehashing the basics of dealing with exponents

Struggling to put two equations together effectively again. I have my income equation: $$ y=zk^\alpha $$ And I'm trying to plug it into my marginal product of capital $$ MPK=\alpha z k^{\alpha-1} ...
3
votes
3answers
227 views

Which is greater: $1000^{1000}$ or $1001^{999}$

Question: Find the greater number: $1000^{1000}$ or $1001^{999}$ My Attempt: I know that: $(a+b)^n \geq a^n + a^{n-1}bn$. Thus, $(1+999)^{1000} \geq 999001$ And $(1+1000)^{999} \geq ...
1
vote
1answer
45 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 ...
0
votes
2answers
44 views

No real solution to logarithmic equation?

$$e^x + 1 = 2e^{-x}$$ Wolfram Alpha claims no real solution and my text book claims the solution $x=0$. Why can't I simply multiply each side by $e^x$: $$e^{2x} + e^x = 2$$ $$\ln(e^{2x}) + ...
1
vote
2answers
25 views

Exponential Growth Rates

So if you are given two different numbers to determine a growth rate, do you use to largest number compared to the value when x=0. For example the problem I am working on is: Your grandfather ...
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
16 views

Variable Base with Variable as Factor in Exponent, Find Value

I saw a problem recently that looked like this: Assume $w$ and $z$ are positive. If $z^{4w} = 64$, what does $z^{6w}$ equal? And I had absolutely no idea how to even begin attempting this equation. ...
0
votes
2answers
46 views

rational exponents. two differing answers.

This is not homework. Example 3) (d) of section P.4, rational exponents in Algebra and Trigonometry: $$\frac{1}{\sqrt[3]{x^4}} = \frac{1}{x^\frac43} = x^{-4/3}$$ Completely rational. Almost ...
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 ...
0
votes
2answers
50 views

Simple question on exponentiation

I know this one is trivial, but I was wondering: if I have something like $$a^{b^c}$$ then i know that it should be read as $$a^{\left(b^c\right)}$$ if no other parenthesis is present. Question: if ...
0
votes
2answers
49 views

If $\sqrt{x+y}+\sqrt{y+z}=\sqrt{x+z}$, then $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=?$

If $\sqrt{x+y}+\sqrt{y+z}=\sqrt{x+z}$, then $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=?$ I really am stumped on this problem. I squared the first equation and found that $-y = \sqrt{(x+y)(y+z)}$. So ...
21
votes
8answers
2k views

How does the exponent of a function effect the result?

The $x^{2/2}$ can be represented by these ways: $$\begin{align} x^{2\over2}=\sqrt{x^2} = |x|\\ \end{align} $$ And $$\begin{align} x^{2\over2}=x^{1} = x\\ \end{align} $$ Which one is correct? And what ...
0
votes
3answers
69 views

Intricate exponential equation

This is the question: $$ \frac{(2^{3n+4})(8^{2n})(4^{n+1})}{(2^{n+5})(4^{8+n})} = 2 $$ I've tried several times but I can't get the answer by working out.I know $n =2$, can someone please give me some ...
0
votes
1answer
19 views

Simplify this indices?

Simplify this: $6a^3 * {a^{-5}\over2}$ I got $6a^3 * {1\over2a^5}$ What should I do next? please explain with steps.
41
votes
10answers
2k views

What is exponentiation?

Is there an intuitive definition of exponentiation? In elementary school, we learned that $$ a^b = a \cdot a \cdot a \cdot a \cdots (b\ \textrm{ times}) $$ where $b$ is an integer. Then later on ...
2
votes
7answers
129 views

for $n$ an integer, why is $n^0=1$ ??

This is so going to cost me.... I was wondering why for any integer $n$: $n^0 =1$. Perhaps It's because $n$ is a round number and if $m$ is a non negative integer, also round then: $$n^m = 1 \cdot ...
2
votes
2answers
588 views

Exponent rule and square roots?

For some $x$, $\sqrt{x^2} = |x|$ However, for $x= -1$. $\sqrt{(-1)^2} = (-1^2)^{1/2} = (-1)^{2/2} = (-1)^1 = -1$ Isn't this paradoxical?
5
votes
2answers
109 views

Prove that $\sqrt{8}=1+\dfrac34+\dfrac{3\cdot5}{4\cdot8}+\dfrac{3\cdot5\cdot7}{4\cdot8\cdot12}+\ldots$

Prove that $\sqrt{8}=1+\dfrac34+\dfrac{3\cdot5}{4\cdot8}+\dfrac{3\cdot5\cdot7}{4\cdot8\cdot12}+\ldots$ My work: $\sqrt8=\bigg(1-\dfrac12\bigg)^{-\frac32}$ Now, I suppose there is some "binomial ...
0
votes
3answers
49 views

How do I compute the individual terms of a polynomial to the power of -1?

If my polynomial $p$ is: $x+1$, obviously $p^{-1} = \frac{1}{x+1}$. Is it possible for me to split $\frac{1}{x+1}$ into a sum of two terms? In other words, is there an algorithm to write $p^{-1}$ as ...
0
votes
3answers
62 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 ...
0
votes
4answers
71 views

How to compute the exponent?

So I have $a^n = b$. When I know $a$ and $b$, how can I find $n$? Thanks in advance!
2
votes
2answers
84 views

Solution for $4^{2x+1}-3^{3x+1}=4^{2x+3}-3^{3x+2}$

Im trying to get a solution for: $4^{2x+1}-3^{3x+1}=4^{2x+3}-3^{3x+2}$ My main problem is thati dont now how to combine this potencys! Ive also thought about another function that would bring me ...
0
votes
1answer
88 views

Exponential equation+derivative

I saw here on math.stackexchange.com an equation which has very nice solutions (by solutions I mean a proof): $3^x+28^x=8^x+27^x$, where $x$ is a real number. However, I think there must be an ...
1
vote
1answer
99 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
4answers
83 views

Calculate approximately the expression $A = 5^{1/2} . 5^{1/4} . 5^{1/8}…$

Calculate approximately the expression $A = 5^{1/2} \cdot 5^{1/4} \cdot 5^{1/8}\cdot\ldots$ My books says to says to do this: $ 5^{1/4} \cdot 5^{1/8} \cdot 5^{1/16}\cdot\ldots = A $ Then $ A = ...
2
votes
2answers
93 views

Definitive answer to existence of real exponents

Well, I've been searching through this fórum and I know this question has been answered many times. But the answers I see, are kinda circular (I think). Let's start by the natural case. Natural case ...
0
votes
1answer
37 views

Prove that $\forall \, a,b \in \mathbb{N}- \{0,1\}\,\, \wedge \,\,a<b \,\, ; \,\, a^{1/a} > b^{1/b}$

Prove that $\forall \, a,b \in \mathbb{N}- \{0,1\}\,\, \wedge \,\,a<b \,\, ; $ $$\,\, a^{1/a} > b^{1/b}$$ I need some tip to start it. Thank you.
3
votes
4answers
101 views

Calculating $\log_7 125$

So the problem asks to calculate $\log_7 125$. It's multiple choice and the options are $2.48$ $4.75$ $1.77$ $2.09$ Given that $7^2 = 49$ and $7^3 = 343$, the answer must be either option 1 or 4, ...
1
vote
3answers
55 views

Exponential Equations

I solved this , but I am not sure if I did in the right way. $$2^{2x + 1} - 2^{x + 2} + 8 = 0$$ $$2^{x + 2} - 2^{2x + 2} = 8$$ $$\log_22^{x + 2} - \log_22^{2x + 2} = \log_28$$ $$x + 2- 2x - 2 = ...
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
47 views

Solve $3^{1/4} \cdot 9^{-5/8}$

I don't understand how to solve $3^{1/4} \cdot 9^{-5/8}$. Help please? I have tried many different things, but they're not working. Once I plug the problem into a math equation solver, the answer ...
0
votes
1answer
737 views

variable with negative exponent in the denominator moved to nominator and vice versa

The top and bottom of the fraction both contain negative exponents. Since $c^{-3}$ on the bottom has a negative exponent, it is moved to the top of the fraction (numerator). Since the $d^{-3}$ on the ...
0
votes
2answers
209 views

How to find $x^4+y^4+z^4$ from equation?

Please help me. There are equations: $x+y+z=3, x^2+y^2+z^2=5$ and $x^3+y^3+z^3=7$. The question: what is the result of $x^4+y^4+z^4$? Ive tried to merge the equation and result in desperado. :( ...
2
votes
1answer
78 views

Trouble solving polynomial equation with exponent

I'm having trouble solving this equation.It looks simple, but I just can't find the answer.Can someone help me? $$9x^4-13x^2+4 = 0$$
0
votes
1answer
29 views

Question about basic exponential/logarithm properties

Solve for $k$: $$e^{k/2}=a$$ Solution: $$e^{2k}=a$$ $$ k/2 = \mathbf{ln}a$$ $$ k=2\mathbf{ln}a$$ $$= \mathbf{ln}a^2$$ My question is: why does $2\mathbf{ln}a = \mathbf{ln}a^2$? Why can you ...
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
0
votes
1answer
106 views

Solve $a=x^n$ , $b=(x+1)^n$ for $x,n$

$$a=x^n~,~b=(x+1)^n$$ Just trying to solve these for $x$ and $n$ . For some reason WolframAlpha gives me a blank screen? Much thanks for any help.
1
vote
2answers
71 views

Math question from the GMATprep

If $xy=1$ what is the value of: $2^{(x+y)^2}/2^{(x-y)^2}$ A 1 B 2 C 4 D 16 E 19 $(x+y)^2/(x-y)^2$ because $2$ just cancels out from numerator and denominator, ...
2
votes
1answer
105 views

Solve a system of equations with $3$ unknown powers?

I'm trying to solve this, knowing $X$ $$\begin{cases}1^x*2^y*3^z=X\\x+y+z=13\end{cases}$$ So for example, if $X=2048$ , we have $$x=2\\y=11\\z=0$$ I barely have memories from high school ...
2
votes
1answer
137 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 ...
3
votes
3answers
77 views

Can a fourth-order equation be solved like a quadratic equation?

I was asked to find the zeros of $y = x^4 + 5x^2 +6$. I tried to turn this into a quadratic to factor it as follows: $y = x^4 + 5x^2 +6 = {(x^2)}^2 + 5{(x^2)}^1 + 6$ Put another way: Let $t = ...
2
votes
4answers
211 views

How to find $\sqrt{1+{4\over x}+{4\over x^2} }$?

If $$abx^2 = (a-b)^2(x+1)$$ then what is $$\sqrt{1+{4\over x}+{4\over x^2} }$$ (A) $a+b \over a-b$ (B)$a-b\over a+b$ (C) $a^2+ab$ (d) None EDIT: What I've done is this: ...