Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [exponentiation]

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

Exponentiation is a mathematical operation which produces a power $a^n$ from a base $a$ and an exponent $n$. The objects involved are usually numbers, but the procedure can be generalized to matrices, elements in algebraic structures, sets, etc.

4326 questions
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solving the equation$(((e^x)^{e^x})^{e^x})^{...}=2$ for x

Let $ \left(x_n\right)_{n = 1}^\infty $ be a sequence of real numbers defined by the initial value $ x_1 = e^x $ for some $ x \in \mathbb{R} $ and the relationship $ x_{n+1} = e^{x^{x_n}} $, such that $ x_2 = e^{x^{e^{x}}} $ and so forth. Hence,…
user82528
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How is $0^x$ defined (seriously)?

Normally, $a^b$ is defined as $e^{b \ln(a)}$ and $e^x$ is defined by its Maclaurin Series, and $\ln(x)$ is defined as the real inverse of $e^x$. If we try this with $0^x$, we get $e^{x \ln(0)}$, which is obviously incorrect. Of course we can just…
wyboo
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$e^{\pi i} = -1$ is this a fact or an assumption?

I think it is a fact but can someone explain why is it true intuitively? I heard a lot of videos on youtube assuming it is the "natural" way of revolving around 0, many other explanations that does not make sense to me. I only need a direct way of…
user37421
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Why don't we extend the naming of powers into higher dimensions?

This is probably a stupid question. If 2^2 is known as "Two squared" and 2^3 is "Two cubed", then why do we stop at 2^3? Why do we not call 2^4 "Two tesseracted", and so on?
Jack Larson
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How to read $( x^{2x^{-1/5}})^{-1}=\frac{1}{25}$

I saw the following problem $$\left( x^{2x^{-\frac{1}{5}}} \right)^{-1}=\frac{1}{25}$$ and I was curious whether I should read it as $$\left( (x^{2x})^{(-\frac{1}{5})} \right)^{-1}$$ or $$\left( x^{(2x^{\frac{1}{5}})} \right)^{-1}$$ to clarify, am I…
Jesse
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Is it possible to rewrite $exp(x)$ with $x$ as the base

Simple question, rewriting $e^x$ with $x$ as the base- or more generally, trying to remove $x$ from the exponent. I don't know if it's possible, as I played around with it for a little while and got nothing. If it is, I suspect it has something to…
MukundKS
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Find $x$ such that $ 2^{16^x} = 16^{2^x} $.

Find $x$ such that $ 2^{16^x} = 16^{2^x}.$ I am a bit confused, what will happen when we expand $16^{2^x}$, will we get $4^{2^{2^x}}$ or $4^{2^{x+1}}$ ?
Raheel
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What is wrong in this calculation

My teacher told me that $\sqrt{x}=x^\frac{1}{2}$, but then why is this incorrect? $$a^2+b^2=c^2$$ $$\sqrt{a^2+b^2}=c$$ $$(a^2+b^2)^\frac{1}{2}=c$$ Multiply in $\frac{1}{2}$ $$a^1+b^1=c$$ $$a+b=c$$ This seems totally correct to me, but we know that…
Johannes
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Solving an equation with $x$ as powers

How would I go about solving $$2^x -2^{x-2}=3 *2^{13}$$Hints please. Thank you.
Koba
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Simultaneous multiple expansion

I am working on algorithm called as Simultaneous multiple exponentiation, I need to understand the mathematical meaning, like from $j=0$ to $k-1$, how, we calculate the value for $G_i$? What does the mathematical expression mean? $For\space\…
Neha
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How to prove that $(a^n)^m = a^{nm}$ for non-integer exponents?

This is a question that has been twinging me for a while. I have been told that $\sqrt{a} = a^{1\over{2}}, a\in\mathbb{R_+}$ and ${1\over{a}} = a^{-a}, a\in\mathbb{R_*}$ were nothing more that notations, not actual properties, and that, therefore,…
user8171079
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Formula to count layers of damascus steel

I have an n layer (forge-welded) steel billet. I fold it i times. The total layer count doubles with each fold. How can i express this with a simple formula? Is it simply: n*(2^i) ? For example 13 layers folded 5 times: 13*(2^5)=416
voices
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What maths rule allows this expression with powers to be rewritten as below?

I have been reading through a programming book and the author asked to calculate the following expression in the program. $$4*5^3+6*5^2+7*5+8$$ I approached this by expanding the expression like so: $$(4*5*5*5)+(6*5*5)+(7*5)+8$$ $$500+150+35+8 =…
Grant Rogers
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Laws of Exponents if base(s) negative

In the case where $a,b>0$ and $x,y \in \mathbb R$, we have Laws of Exponents such as these: $a^xa^y=a^{x+y}$ $(ab)^x=a^xb^x$ $a^{-x}=1/a^x$ $(a^x)^y=a^{xy}$ $a^x/a^y=a^{x-y}$ $(a/b)^x=a^x/b^x$ $(a/b)^{-x}=b^x/a^x$ If instead $a,b<0$, then Which…
user693894
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Are there other powers which follow the rule $a^b = b^a$ than $2^4$?

I was trying to find these powers, but to my disappointment I only found $2^4 = 4^2$. Edit: $a$ must be different to $b$ of course. Is that the only possible setting, and why? If we assume the number and the power can have also decimal values.
mathnewbie
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