# Questions tagged [exponentiation]

Questions about exponentiation, the operation of raising a base $b$ to an exponent $a$ to give $b^a$.

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### Are there natural numbers $x$, $y$, $z$, such that $x^4+y^4-2^{3z}=3$?

Are there natural numbers $x$, $y$, $z$, such that $x^4+y^4-2^{3z}=3$? Justify!! I literally have no idea for this problem. I thought of doing the last digit but it doesnt help at all. I wonder if we ...
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### How to find rational power of two using Newton's method?

I know how to find integer powers of two, $2^x = \prod_{i=1}^x 2$, I have memorized powers of two up to 32nd power of two, and I use bit-shifts to calculate them. For integer powers of other numbers I ...
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### Why doesn't $x^0$ equal the same thing as $x\cdot0$?

If $x\cdot0$ means adding zero $x$s together and $x^0$ means multiplying zero $x$s together then conceptually why aren't they both equal to the same thing?
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### Simply calculate $\frac{ \frac{65536!}{(65536 - 100)! }}{65536^{100}}$

I want to calculate this value exactly: $C=\frac{ \frac{65536!}{(65536 - 100)! }}{65536^{100}}$. Or at least I want to obtain an a convincing lower bound to argue that this value is close to 1. It can ...
1 vote
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### Solve $^y{x} =$ $^x{y}$ over the real numbers

Let $x, y \in \mathbb{R}^{+}$ be such that $x \neq y$ and assume $n \in \mathbb{N}-\{0\}$. Now, referring to the well-known hyperoperation sequence $x[n]y$, we have that $xy=x+y$ and we know that ...
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### Finding a real closed-form solution to a tricky transcendental equation

One of my friends posed the following question with a prize of free bubble tea for anyone who could find a closed-form solution to the problem (not necessarily in terms of elementary functions): What ...
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### Convergence of Variable Base Power Tower $\frac{1}{2}^{\frac{1}{3}^{\frac{1}{4}^{\ldots}}}$

I’m curious if there is a heuristic method that can be used to solve what appears to be an elementary power tower where every increasing power is decreasing by a half integer. Numerical methods ...
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### Bounding exponential function by n^{-\gamma}

Assume that we know $$\exp(-\frac{t^2/2}{np(1-p)+t/3})$$ and our goal is to obtain which value of $t$ we can take such that the above is bounded by $n^{-\gamma}$, for certain constant $\gamma$. The ...
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### Why does the function $e^{ix}$ have a real part, without using the Euler's formula

I would like to intuitively understand why $e^{ix}$ has a real part, if the the function $e^{ix}$ has an imaginary argument. I know that $$e^{ix}=\cos x + i\sin x$$ and I don't need convincing that it ...
1 vote
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### Solve for $x$: $3^{2x}=5^{x-1}$

So I recently got myself an AP Precalculus book (for anyone who wants to know which book, I will have that in the "To Clarify" section at the bottom of this post) and was looking through ...
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### Domain of $\frac{1}{x^{-1}}$

Why WolframAlpha and other online calculators compute $x\in \mathbb{R}-\{0\}$ (and not $\mathbb{R}$) as the domain of $\frac{1}{x^{-1}}$? Since $\frac{1}{x^{-1}}=\frac{1}{\frac{1}{x}}=x$, is not the ...
1 vote
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### Are there any non-trivial solutions of $a^n = b^m + 1$ for natural $a$, $b$ and $n, m \ge 2$

There is an obvious solution $3^2 = 2^3 + 1$. However, I tried to search for solutions in the range $a, b \in [2,100]$ and $\min(n,m) \in [2,1000]$, but did not find any. It is easy to show that $a$ ...
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### What is the resulting precision from an exponent operation?

Say I want to perform exponentiation on some decimal number (like a measured weight) and some integer exponent (like ^2 or ^3). What are the general rules for the resulting precision(significant ...
1 vote