# Tagged Questions

For questions pertaining to power towers: expressions like a^(b^(c^d))), which result from iterated exponentiation. The "hyperoperation" tag may be appropriate, too.

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### Integrate $x$ to the power $x$… to the power $x$… infinitely

This came across my mind, integrating $x$ to the power $x$ infinitely, I couldn't find anything on it. $$\Large \int x^{x^{x^{x\,\cdots}}} \, dx$$ How would you go about this?
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### What is the generating function $G(x,y)$ for powertowers of $x$?

I'm looking for a generating function $G(x,y)$ for powertowers of $x$ such that $G(x,y) = 1 + xy + x^x {y^2 \over 2!} + x^{x^x} {y^3 \over 3!} + x^{x^{x^x}} {y^4 \over 4!} + ...$ Is there any ...
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### Extra root of hyperpower equations [duplicate]

Consider the hyperpower equation $$x^{x^{x^{x...}}}=2$$ We will use the method that Let $y=x^{x^{x^{x...}}}$ so $x^y=x^{x^{x^{x...}}}=2$ and $x^2=2$ to give the solution $x=\sqrt2$ However ...
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### What is the derivative of: $f(x)=x^{2x^{3x^{4x^{5x^{6x^{7x^{.{^{.^{.}}}}}}}}}}$?

I happened to ponder about the differentiation of the following function: $$f(x)=x^{2x^{3x^{4x^{5x^{6x^{7x^{.{^{.^{.}}}}}}}}}}$$ Now, while I do know how to manipulate power towers to a certain extent,...
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### What is the derivative of $x!^{x!^{x!^{x!^{x!^{x!^{x!^{.{^{.^{.}}}}}}}}}}$

What is the derivative of $$x!^{x!^{x!^{x!^{x!^{x!^{x!^{.{^{.^{.}}}}}}}}}}$$ My effort: Let $$g(x)=x!^{x!^{x!^{x!^{x!^{x!^{x!^{.{^{.^{.}}}}}}}}}}=>g(x)=x!^{g(x)}$$ Taking natrual log on both ...
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### Could you solve $x↑^n2=x↑^m2$?

As my title asks, could you solve $x^2=2^x$? But that's the worrisome part, as I noticed $x↑^n 2=x↑^m 2$ and $2↑^p x=2↑^q x$ will always have a solution at $x=2$. However, there is bound to be at ...
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### If $x^{x^4}=4$, then what is the value of $x^{x^2}+x^{x^8}$?

If $x^{x^4}=4$, then what is the value of $x^{x^2}+x^{x^8}$? I tried to simplify it using exponentiation and logs, and even just algebraic manipulation..But I don't know how to do this.
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### Solving for $x$ in $M ^ {M ^ M} = x ^ {1 / (x-1)}$ where $M = 5 ^ {\sqrt{5} / 10}$

Methods used by analogy, for example $x ^ x = 3 ^ 3 \implies x = 3$, Determine the value of $x$ in $$M ^ {M ^ M} = x ^ {1 / (x-1)}$$ if $M = 5 ^ {\sqrt{5} / 10}$.
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### Number Theory : What are the last three digits of $9^{9^{9^9}}?$

I was doing some basic Number Theory problems and came across this problem and was all thumbs : Find the last three digits of $9^{9^{9^9}}$ How would I go about solving this problem? I am a ...
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### How find the number of zeros at the end of the sum $4^{5^6}+6^{5^4}$?

The problem is to find the number of zeros at the end of the sum $4^{5^6}+6^{5^4}$. I tried $2^{2 \cdot 5^6}+3^{5^4} \cdot 2^{5^4}= 2^{5^4} \cdot ( 2^{2 \cdot 5^6 -5^4}+ 3^{5^4} )$.
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### Prove that $2^{2^{\sqrt3}}>10$

With a computer or calculator, it is easy to show that $$2^{2^\sqrt{3}} = 10.000478 \ldots > 10.$$ How can we prove that $2^{2^{\sqrt3}}>10$ without a calculator?
### Summation of $2^{(-2^{n})}$ [duplicate]
By the ratio test, I know that this series convernges: $\sum2^{(-2^{n})}$, in the limit $n$ goes to infinity. Probably to something close to $.8$ (if not equal to $.8$). The problem is, how do I ...