# 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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### Showing the inequality $f(x)=x^{2(1-x)}+(1-x)^{2x}\leq 1$ for $0<x<1$

My proof is not really natural but I think it works. We want to show : Let $0<x<1$ such that then we have : $$f(x)=x^{2(1-x)}+(1-x)^{2x}\leq 1$$ Case $0<x\leq 0.25$ The proof of this ...
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### What is wrong with this argument that $(-2)^{1/4}=4^{1/8}$?

My prof started today's lecture by writing $$(-2)^\frac{1}{4} = (-2)^{(2*\frac{1}{8})} = ((-2)^{2})^{\frac{1}{8}} = 4^{\frac{1}{8}}$$ and asked us whether this was valid or not. However he didn't ...
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### If $n$ is an rational, when is $n^{1/n}$ rational?

Given that $n$ is rational, when is $\sqrt[n]{n}$ rational? We can make a polynomial $x^{n}-n$ whose root is $\sqrt[n]{n}$ and using RRT we can show that there are no rational roots, but in the ...
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### How do you solve $2^{x-1}=\frac{1}{x}$?

$2^{x-1}=\frac{1}{x}$ Clearly by substitute $x=1$ we were able to solve this problem but how do we really solve it using calculus?
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### Solve the inequality $a^{ab}c+b^{bc}a+c^{ca}b\leq 1-2abc$

Let $a,b,c>0$ such that $a+b+c=1$ then we have : $$a^{ab}c+b^{bc}a+c^{ca}b\leq 1-2abc\quad (1)$$ I have a proof : I was thinking for an alternative proof considering by example Young's ...
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### Expansion of square root of a sum

I know that $(a + b)^2$ can be expanded as $(a + b) * (a + b) = a^2 + 2ab + b^2$. Is there an equivalent expansion method for the square root of a sum, that is, $(a + b)^{1/2}$? If there's no method,...
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### Infinite Power Tower approximation: float error? [closed]

Desmos appears to plot it falsely using the $x^y = y$ definition, curving backwards. I've included a 50x exponent for comparison, which suggests no values flowing left in $x$-axis due to float error - ...
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### Prove that for $a>2-2\log(2)$ the equation $e^x= 2x+a$ there are 2 distinct solution [closed]

I have to prove that $e^x= 2x+a$ for $a>2-2\log(2)$ there are 2 distinct solution. I've thought using Bolzano's theorem but I can't understand where to start, thanks in advance. Edit: the solution ...
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### $e$ and $\ln$ : how to derive two equivalent equations

When solving the equation $$150 = 160 - 40 e^{-t/20}$$ I come to a solution that seems natural to me as follows: \begin{align*} .25 &= e^{-t/20}\\ \ln(.25) &= -t/20\\ t &= -20 \ln(.25)\\ &...
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### Show that $3^{22}-2^{20}$ is divisible by $7$

I have this one question which after some hours of thinking I can't seem to be getting anywhere. The question reads: Show that $3^{22}-2^{20}$ is divisible by $7$. Now, after using a calculator I know ...