# Tagged Questions

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### From an expression raised in a power of 2 to an expression raised in the power or 10

Is there a simple/"easy" way to convert a big number from a power of $2$ to a power of $10$ equivalent. Example: I had $2^{127}\cdot 1.9999999$ which I did the multiplication got the result and from ...
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### Solve the equation: $1+2^x+4^x+8^x+16^x+32^x=3(1+2^x+4^x)$

I am doing some math repetition and am a bit stuck on this exercise: Solve the equation: $1+2^x+4^x+8^x+16^x+32^x=3(1+2^x+4^x)$. Now, this is a geometric sum on both the $LHS$ and $RHS$, which I ...
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### How to divide $6^{4/6}$ by $6^{6/8}$?

How do you resolve this: base 6 and exponent 4/6 divided by base 6 and exponent 6/8?
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### Why is the result of $-2^2 = -4$ but $(-2)^2 =4$?

I am really new into math, why is $-2^2 = -4$ and $(-2)^2 = 4$?
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### How does $2^n + 2^n = 2^{n+1}$?

What property of exponents can be used to show that $$2^n + 2^n = 2^{n+1}$$ Does this work for all constants raised to a variable exponent?
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### Solve for $x$ in the equation [closed]

Please help me to solve for x using maybe logarithm or exponential rules (or both) $$5^x=2 \cdot 3^x$$
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### generalized way of finding pair solutions of an equation

I want to find out pair solutions of this equation: $$x^{2}-79y^{2}=1$$ This is a hyperbola equation. I sketched its graph, but that didn't help me. I think the square from (form?) of $x$ and $y$ is ...
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### Solution for this Logarithmic Equation

Recently I was going through a problem from the book Problems in Mathematics - *V Govorov & P Dybov* . $$(x-2)^{\log^2(x-2)+\log(x-2)^5-12}=10^2\log(x-2)$$ I tried solving by first considering ...
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### Solving $e^{4x}+3e^{2x}-28=0$

How to solve this equation: $$e^{4x}+3e^{2x}-28=0$$ I don't know how to solve this problem. I read over another example, $e^{2x}-2e^x-8=0,$ and it said that $e^{2x}$ is $e$ to the $x$ squared, ...
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### How can I simplify the expression $\frac{\sqrt[5]{x^2}}{x^2}$?

$$\dfrac{\sqrt[5]{x^2}}{x^2}$$ I'm doing a summer math packet for calculus. I need to simplify the above. I think I may know the answer, but I'm not sure. Can someone help me, please?
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### Calculate $\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$

I'm an eight-grader and I need help to answer this math problem. Problem: Calculate $$\frac{1}{5^1}+\frac{3}{5^3}+\frac{5}{5^5}+\frac{7}{5^7}+\frac{9}{5^9}+\cdots$$ This one is very hard for me. It ...
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### 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: ...
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### 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? $3^x=2^x+2$
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### 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 ...
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### Finding the exponent of $2$ such that $x \cdot 2^a$ is as close to $1$ as possible

How do I find an exponent of $2$ that when multiplied with another number would bring the result closest to the positive side $1$? Like this: $y = x \cdot 2^a$, where $y\ge 1$ has to be as small as ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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: $$\frac{16^{x+1}+4^{2x}}{2^{x-3}8^{x+2}} \text{ is ? }$$
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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 ... 2answers 737 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? 2answers 111 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 ... 3answers 50 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 ... 3answers 65 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 ... 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! 2answers 85 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 ... 1answer 90 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 ... 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 ... 4answers 84 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 = ... 2answers 97 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 ... 1answer 38 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}$$I need some tip to start it. Thank you. 4answers 105 views ### Calculating \log_7 125 So the problem asks to calculate \log_7 125. It's multiple choice and the options are 2.48 4.75 1.77 2.09 Given that 7^2 = 49 and 7^3 = 343, the answer must be either option 1 or 4, ... 3answers 57 views ### Exponential Equations I solved this , but I am not sure if I did in the right way.$$2^{2x + 1} - 2^{x + 2} + 8 = 02^{x + 2} - 2^{2x + 2} = 8\log_22^{x + 2} - \log_22^{2x + 2} = \log_28x + 2- 2x - 2 = ...
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 ...