Questions about exponentiation

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Elementary Analysis, 3rd root question

Prove that $\forall a \in \mathbb{R}$ there is a unique solution to $x^3 = a$ Prove that $\forall x,a \in \mathbb{R}$ $$(x^{1/3}-a^{1/3})(x^{1/3})^2 + a^{1/3} x^{1/3} + ((a^{1/3})^2)=x-a$$ Prove ...
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1answer
38 views

Is there a such a thing as a “log root” or perhaps a “power root”?

We all know what a square root is. It's the number that for any $x$, you can calculate $n * n = x$. Is there an equivalent function to determine for any $x$ that you can calculate $n ^ n = x$? How ...
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Golden Ratio Sandbox

This might be a little long so please bear with me. The Golden Ratio $\phi$ is defined as the single positive root of the polynomial $p(t) = t^2 - t - 1$. One can think of it as a line divided into ...
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2answers
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derivative of $\frac{2}{3}x^{3-e}$

Find the derivative:$\;\;\;\;\;\;\dfrac{2}{3}x^{3-e}$ I am not sure how to solve this problem. My try: $\ln y=\dfrac{2}{3}(3-e)\ln x$ $\dfrac{1}{y}\times y\;'=\dfrac{2}{3}(3-e)\dfrac{1}{x}$ ...
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What does it mean to have an irrational/imaginary exponent and is there a way to calculate the latter?

In exponentiation, we are told that raising something to an integral power (n, say) means multiplying it with itself a total of n times, if n is non-negative. And we also learn fairly early on that ...
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3answers
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Evaluating the value of exponential expression

What is the value of: $\frac{2a}{a^{x-y}-1}+\frac{2a}{a^{y-x}-1}$ I tried this: $(\frac{a^{x-y}-1}{2a})^{-1}+(\frac{a^{y-x}-1}{2a})^{-1}$ $\frac{({a^{x-y}-1})^{-1}+({a^{y-x}-1})^{-1}}{2a^{-1}}$ ...
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2answers
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Difficulty understanding addition of exponents

If you rewrite $(3^{12})(3^{-12})$ in the form $3^n$ what does it equal? What is the intuition behind it? Do exponents cancel each other out so it is just $3$? or do the negatives cancel out $((-x) ...
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32 views

derivative of $e^{\ln x^2}-3x^7$

$$e^{\ln x^2}-3x^7$$ The first term: $=e^v$ $v=\ln x^2=u^2$ $v\;'=2uu\;'=(2\ln x)\dfrac{1}{x}=\dfrac{2\ln x}{x}$ $\dfrac{e^{\ln x^2}2\ln x}{x} +21x^{-8}$ How do I simplify further? I don't ...
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How to make powers work right on a casio calculator?

I pressed some random buttons and it worked fine,but now its not working again. for example my calculator shows that -6^2 is - 36 and not 36, anyone know how to fix it??? My clculator model is casio ...
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Derivative of $e^\sqrt{4x+4}$

$$f(x)=e^\sqrt{4x+4}$$ $f(x)=e^u$ $u=\sqrt{4x+4}=(4x+4)^{1/2}$ $u\;'=\dfrac{1}{2}(4x+4)^{-1/2}=\dfrac{1}{2\sqrt{4x+4}}$ I don't know how to proceed from here. Thanks.
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1answer
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The sum of powers of two

While doing a proof of correctness for an algorithm, I ran into a roadblock with a smaller proof. The problem in words: A set with all elements which are a power of 2, and the sum of the set is ...
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3answers
21 views

Calculating sum of consecutive powers of a number

Here is my problem, I want to compute the $$\sum_{i=0}^n P^i : P\in ℤ_{>1}$$ I know I can implement it using an easy recursive function, but since I want to use the formula in a spreadsheet, is ...
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1answer
32 views

What is the value of x? Related to Indices.

Just some days ago I appeared for a maths exam. In that exam there was a question related to Indices which I was not able to solve. After the exam I even tried solving it in the home next 2 days but ...
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2answers
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Evaluating a limit as $x \to -\infty$ of a power of a rational function

Sorry for the weird title, I don't know how to put the equation on the title. $$\lim_{x\to-\infty}\left(\frac{1-x^3}{x^2+7x}\right)^5$$ Ok I divided inside the parenthesis by $x^2$, but now I am ...
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3answers
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Is it true that $0^0$ is undefined? Why or why not? [duplicate]

Is it possible for zero to the power of zero to be undefined? Is there a good reason if it IS undefined? If yes, I hope there is! This question is different because I'm trying to figure out if $0^0$ ...
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1answer
46 views

Right way to solve for $\frac{2^{900}*7^{898}}{14^{897}}$

As a sequel to my question How to solve $0.5^{1200}\times (2^{1204})$? : $\frac{2^{900}*7^{898}}{14^{897}}$ Will I first solve the upper raw like did in previous question and then anwser $14^{897}$
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4answers
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How to solve $0.5^{1200}\times (2^{1204})$?

I've been struggling with this one. I know that the anwser is $16$, but how do I solve this on paper? $0.5^{1200}\times 2^{1204}$ I know that this has something to do with first subtracting the ...
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3answers
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Calculating the value with large exponents

I'm trying to solve $2014^{2015}$ $\pmod {11}$, is there a trick or tip to break the problem down to make it easier to solve?
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Why is $3^{(x-5)} + 3^{(x-7)} + 3^{(x-9)} = 91$?

So far I think that this is somehow related to that $(x-7) - (x-5) = (x-9) - (x-7) = 2$, but is it ? What steps do you take to add $3^{x-5} + 3^{x-7} + 3^{x-9}$ up ? Thank you!
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2answers
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Chaining Exponent Rules Together

I'm having trouble understanding why the following property is true and want to make sense of it before going ahead and use it in my proof by induction: $$2^{2^n}=2^{2^{n-1}}\times 2^{2^{n-1}}=\left( ...
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1answer
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Power function and involution $f(x) = x^a$

For power functions we have a variable $x$ and a constant $a$; we get that $f(x) = x^a$. Find all involutions for $f(x)$. I started out with basic functions such as $f_1(x) = f_1^{-1}(x) = x^1$ and ...
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Is there a shortcut for raising 2 to the power of a number (e.g. $2^{27}$)?

In networking, when dealing with subnetting, you convert the net mask to binary and count the number of ones (for the example in the question there would be $27$ $1$'s) and to figure out how many ...
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3answers
62 views

How can we prove that the square root of any number is equal to the statment given below

Is there any theorem that can explain this $\sqrt[n]{x}= x^{1/n} $, is there any practical example of $ x^{1/n} $, 1/n times of a number.
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Relationship of powers of Phi to Lucas Numbers

I was watching a Numberphile and the interviewee was explaining various attributes of Lucas Numbers and he made the statement about creating a sequence by starting with the Golden Ratio and raising it ...
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1answer
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Confusion in “mask bits”

This is the question: By definition of the IEEE754 standard, 32-bit floating point numbers are represented as follows: S (1 bit) E (8 bits) M (23 bits) S: Sign bit E: Exponent M: ...
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2answers
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Simplify an expression involving indices

One of my friends asked me this question: Simplify $$\frac{50^{3x-1} 10^{2-3x}}{250^{3x+1}}$$ I've been thinking about the question for more than a day. I've looked through my teacher's notes but ...
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2answers
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Solving equations with fractional indices

How would I go about solving an equation like this? $3^{4/3}b^{5/3} - b^3 = 1$ I thought about rearranging to get $3^{4/5}b = (1 + b^3)^{3/5}$, but that didn't seem to lead anywhere as I couldn't ...
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Exponents with the power being a negative/decimal?

How would I do the equation $b^{n}$ Or $b^{-n}$ Where $b$ is the base, and $n$ in a negative/decimal?
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What's the inverse operation of exponents?

You know, like addition is the inverse operation of subtraction, vice versa, multiplication is the inverse of division, vice versa , square is the inverse of square root, vice versa. What's the ...
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1answer
41 views

Irrational power of root

Let $a$ and $b$ be rational numbers, such that $\sqrt{a}$ and $\sqrt{b}$ are irrational. Can $\sqrt{a}^\sqrt{b}$ be rational? I found examples, where the irrational power of an irrational number is ...
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1answer
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Simplifying $2^\sqrt{\log x}$

Can this expression be simplified? $$2^\sqrt{\log x}$$ Thank you
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Power of very big numbers

So, I was solving a question, and I came across this. If I have, x=a^b, and If I want to calculate the last digit of x, then it ...
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4answers
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$4^{x}+2^{x+1}=18$ Please help me solve?

I tried using logs on both sides or tried treating it as a quadratic but didn't manage to simplify it, Help?:D
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2answers
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I need to solve this equation [do my homework]. [closed]

Problem for my university math class 1st semester Thanks a lot!
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exponential equation (solve for x)

We started review in my calculus class and I have mostly forgotten everything about exponential equations. $$ e^{-x} * (3e^{2x}-(5/4))^{1/2} = e^x $$ Would x just equal 0, when -x+x=0 (two sides of ...
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is there any solution for $x^2 +x + 2 = e^x$ by using algebra?

I know this can be solved by numerical methods but I would like to know whether this can be solved using logs or something similar. Thanks
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3answers
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Matrix exponential: Formal notation for power series? Or, more?

For a square matrix $A$, I'm already used to see and use: $$\sum_{n=0}^{\infty} \frac{A^n}{n!} = \lim_{n \to \infty} \left(I + \frac{A}{n}\right)^n = e^A$$ Which means a matrix $A$ is just like some ...
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2answers
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Negative Number raised to fractional power

How would you solve a negative number raised to a fraction a/b if b is odd and a is evem? Ignoring imaginary numbers i.e $(-1)^\frac23$ Calculator returns an error $(-1)^\frac 13 (-1)^\frac 13$ = ...
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Can $x^2 +x + 2 = 10^x$ be solved using algebra?

I know this can be solved by numerical methods but I would like to know whether this can be solved using logs or something similar. Thanks
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2answers
28 views

Exponential equations with variables on both sides

I have the following: $$8^{3x+4} = 5^{4x-2}$$ How would I solve this? I tried this: $$(3x+4)\log 8 = (4x-2)\log 5$$ but have no idea where to go from there. Thank you!
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1answer
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calculating a decrease at constant instantaneous rate

If we have a value e.g. $$ B = 20000 $$ and it decreases at a constant instantaneous rate of say $$ -1.1*10^{-2} $$ per unit time. What would B look like over say 300 time units, and how do we ...
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1answer
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How to find the common base of terms in an expression?

I'm teaching myself basic algebra from a book and am stuck on a question. In the current section it is about expressing numbers as powers of the same base. So $9$ maybe expressed as $3^2$. Another ...
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1answer
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Plotting nth root of x against nx on a graph.

I've spent the last two days trying to figure this out. What I'm trying to do is rearrange this: $$ x^n = \frac xn $$ to make n the subject, to allow me to plot on a graph, with n being the ...
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1answer
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Sign of Fractional Exponents

When calculating a number to a fractional exponent or fractional nth root, in what cases is there both a positive and negative solution as opposed to just a positive or just a negative solution?
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How easy is it to prove that if $2^x,3^x, 5^x, 7^x, 11^x … $ are all integers then $x$ is an integer as well?

How easy is it to prove that if $2^x,3^x, 5^x, 7^x, 11^x ... $ are integers then $x$ is an integer as well? I have read the definition of the exponent functions as given in my calculus text, and the ...
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1answer
32 views

exponentials with different base

What rule can I use when solving exponentials like this one $\frac {2^6 \cdot 5^8 \cdot 3}{100^3}$ I know how to solve exponentials when the bas number is the same with these formulas $x^m \cdot x^n ...
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2answers
47 views

Unit Quaternion to a Scalar Power

I'm trying to modify a physics engine for efficiency. Currently, as objects move around the world, their orientation (a quaternion) is updated every frame, by multiplying by the rotation (another ...
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How to prove that $b^{x+y} = b^x b^y$ using this approach?

Fix $b>1$. If $m$, $n$, $p$, $q$ are integers, $n > 0$, $q > 0$, $r = m/n = p/q$, then I can prove that $(b^m)^{1/n} = (b^p)^{1/q}$. Hence it makes sense to define $b^r = (b^m)^{1/n}$. I ...
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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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4answers
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Why is a Constant added to front?

I made the differential equation : $$dQ = (-1/100)2Q dt$$ I separate it and get: $\int_a^b x (dQ/Q) = \int_a^b x (-2/100)dt$ this leads me to: $\log(|Q|) = (-t/50) + C$ I simplify that to $Q = ...