# 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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### Why does zero raised to any positive power equal zero? [closed]

What if you raised 0 to the power of 2?
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### How to prove solutions of $2(2^x- 1) x^2 + (2^{x^2}-2)x = 2^{x+1} -2$?

The original question wanted the real solutions to $$2(2^x- 1) x^2 + (2^{x^2}-2)x = 2^{x+1} -2 .$$ I graphed this equation and worked out the trivial roots $x=-1$ and $x=1$, and I could not find any ...
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### Why is $\exp(n/2 * (\log 2)^2) \approx 1.27^n$

I saw this approximation made in https://stats.stackexchange.com/questions/473496/infinite-coin-toss-probability by the accepted answer. What inspired this approximation?
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### Why is $f(x)$ undefined for negative $x$ on Maple and my calculator but defined on WolframAlpha?

$f(x):=x^{4/5}\cdot (x-4)^2$ Maple gives me the graph: WolframAlpha gives this graph and my calculator gives Error 2 if I plug in $(-1)^{4/5}$. My original intuition would be to think that the graph ...
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### If $\displaystyle x^{x^9}=\sqrt{3^{\sqrt{3}}}$ and $\displaystyle y=x^{\left(\frac{1}{y^{y^x}}\right)}$, determinate the value of $y^{3x}$.

If $\displaystyle x^{x^9}=\sqrt{3^{\sqrt{3}}}$ and $\displaystyle y=x^{\left(\frac{1}{y^{y^x}}\right)}$, determinate the value of $y^{3x}$. My try It is easy to see that if we raise the first equation ...
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### What is the proper convention regarding the order of operations of a fractional exponent and/or the simplification of it?

Specifically, consider the example $\sqrt[4]{x^2}$. The answer of course would be $\sqrt{|x|}$ since the x is squared first. However if converted to the exponential fraction of $x^{2/4}$, you lose the ...
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### $\lim f(t)^{g(t)} = 0^0 = 1$ if $f,g$ are analytic?

Wikipedia states that if $f,g$ are real analytic near $c$, and $f,g \to 0$ as $t \to c$, then along any path for which $f>0$ one has $f(t)^{g(t)} \to 1$. The proof Wikipedia cites is in German so I ...
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### Converting a radical to a fractional exponent

I want to understand how to convert a radical to a fractional exponent. Given the following equation: $\sqrt[3]{(x)^6\cdot x^9}=\sqrt[3]{x^{24}\cdot x^9}=\sqrt[3]{x^{33}}=x^{\frac{33}3}=x^{11}$ How ...
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### Find all nonnegative integer solutions to $2^a +3^b +5^c =n!$

Question: Find all nonnegative integer solutions to $2^a +3^b +5^c =n!$. My work so far: I realize that if $n$ is greater than $2, 3$, or $5$, then $3^b+5^c$ has to be divisible by $2$, $2^a+5^c$ has ...
### Compound interest - relationship between $\frac{r}{n}$ and $r$?
The compound interest formula $A=P\left(1+\frac{r}{n}\right)^{nt}$ is usually used in examples where you are given a nominal annual rate and then calculate the accrued amount, where $\frac{r}{n}$ is ...