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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Representing positive integers as floor of integer powers of real number.

Does there exist real $a$ such that for every positive integer $c$ there exists integer $b$ such that $\lfloor{a^b}\rfloor = c$?
user4201961
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The Exponential Property

Prove true or false for the statement: every $x \in \mathbb{R}$, holds $x^{\frac{6}{2}} = x^3$ The habit of what we did everyday when facing exponential forms like this creates confusion to prove whether it is true and holds for every real numbers…
Shane Dizzy Sukardy
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How can we represent any number as a series of exponents?

Say I have a positive integer that is one-thousand digits long. What math could I use to represent this number as a series of exponents in a significantly shorter form than the original number? The number is rather random. For example, $(4^5)^6$…
Jason
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All such $m\neq n$ such that $n^m = m^n$

I'm curious as to how many positive integers $m\neq n$ exist such that $n^m = m^n$. Is $n = 2$, $m=4$ the only case? I've plotted two surfaces (one surface represents $n^m$ where the other represents $m^n$) on a log-scale. Where they intersect is…
jonem
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Wrong basic exponential rule: $(a^b)^c\neq a^{bc}$

I've been searching about exponential rules on google and the first three results was these ones: Link 1 Link 2 Link 3 I didn't understand why they all say: $(a^b)^c=a^{bc}$. This is wrong, see for example: $((-2)^{2})^{1/2}=2\neq (-2)^1$. Why does…
user42912
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Proof that $2^{2^x}$ ends in 6

So I just checked every number of the form $2^{2^x}$ up to $2^{8192}$ and they all end in $6$. Can someone formally prove that this will be true for all $x$?
Mathime
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What is $25^k$ + $5^k$

This is an extremely simple problem, but I can't find an example anywhere for some reason. I know that $30^k$ is not correct But I have no idea what else makes sense.
123
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How to calculate 10^ decimal power without a calculator?

I need to know how to calculate 10^ a decimal power, like 10^-7.4, without a calculator, in as simple a way as possible, since I will be doing questions which only allow me about a minute to a minute and a half each. Does anyone know a good…
Crystal
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Exponential Properties

Here are my steps: $e^{{2\pi i}/100} = (e^{\pi i})^{{2/100}} = ((-1)^2)^{1/100} = 1^{1/100} = 1$. I'm not sure if the normal rules of exponents apply like this if the power is complex.
Tim
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Comparing large exponents

Without calculator, I have to determine which of the following is larger: $2^{350}$ or $5^{150}$ I know only very basic exponential laws, and haven't covered logarithms yet. Tried various algebraic simplification methods but had no luck. Any help is…
StopReadingThisUsername
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What's the name for this mathematical device used by programmers?

So a friend is trying to figure out what this is called so we can read more about it. The concept is/was used by database designers, who needed a compact way to store a list of selected options as a single numerical value, where that number would…
smohyee
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Why is $n^0 = 1$?

Why is any number to the zeroeth power equal to 1? I would think it would be equal to zero, since nothing multiplied by nothing is, well, I would think 0. But it is 1? Examples: $(-5)^0 = 1$; $0^0 = 1$; $5^0 = 1$;
Evorlor
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Why x by x is NOT equal to squared x within exponents?

Normally you can write $x*x=x^2$ But if you are operating within exponents, $a^{x*x} \neq a^{x^{2}}$ as the latter is equal to $a^{2x}$. Is it a problem of notation ? [Edited] Thank you to having helped me to solve this problem. It is indeed a…
Antonello
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Calculations using googolplexes

How can I calculate $\dfrac{10^{10^{100 }}}{ 10^{10^{70}}}$? I have tried using logs ie: $$\frac{10^{10^{100}}}{10^{10^{70}}}$$ $$=\frac{(100\times \ln(10)) \times \ln(10)}{(70\times \ln(10)) \times \ln(10)}$$ $$=\frac{10}{7}$$ which looks incorrect…
Jayprakash
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Negative Number raised to fractional power

How would you solve a negative number raised to a fraction a/b if b is odd and a is evem? Ignoring imaginary numbers i.e $(-1)^\frac23$ Calculator returns an error $(-1)^\frac 13 (-1)^\frac 13$ = -1.-1 = 1 (By law of indices) or $(-1^\frac13 )^2$…
aveiur
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