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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How do you compute negative numbers to fractional powers?

My teachers have gone over rules for dealing with fractional exponents. I was just wondering how someone would compute say: $$(-5)^{2/3}$$ I have tried a couple ways to simplify this and I am not sure if the number stays negative or turns into a…
Kot
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How to calculate a decimal power of a number

I wish to calculate a power like $$2.14 ^ {2.14}$$ When I ask my calculator to do it, I just get an answer, but I want to see the calculation. So my question is, how to calculate this with a pen, paper and a bunch of brains.
Mixxiphoid
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How does an exponent work when it's less than one?

I'm rather familiar with exponents, I know that $y^x = y_1 \cdot y_2 \cdot y_3 .... y_x$, but what if the exponent is less than one, how would that work? I put in my computer $25^{1/2}$ anyway, expecting it to give me an error, and I got an…
Flostin
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Convert from high exponent of base $10$ to base $2$.

Is there an efficient way to convert from a high exponent of base $10$, to base $2$? Both in exponent notation. Here's an example: If I have a number that's $10^5$ or even $10^{100}$, and I wanted to convert that to base $2$, exponent notation, how…
user2200321
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$5^{x}+2^{y}=2^{x}+5^{y} =\frac{7}{10}$ Work out the values of $\frac{1}{x+y}$

$5^{x}+2^{y}=2^{x}+5^{y} =\frac{7}{10}$ Work out the values of $\frac{1}{x+y}$
tianzhidaosunyouyu
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Why does zero raised to the power of negative one equal infinity?

I had the question of $0^{-1}$ on a math test and I naturally assumed that this evaluates to zero, but from what I have seen from various sources it is equal to infinity which I do not quite understand. I would sooner believe that this it is just…
perrothedog
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10 to the power of 3.5: $10^{3.5}$

So $10^3 = 10\times 10\times 10 = 1000$, this is really easy to understand. But what about: $\,10^{3.5}\,?\,$ My logic would suggest this was $10\times 10 \times 10\times 5 = 5000,\;$ but the calculator says it's 3162.27... Can someone illustrate…
BjarkeCK
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Why are the first few powers of $2^{10}$ a little more than those of 1000?

See the complete list here: http://en.wikipedia.org/wiki/Power_of_two#Powers_of_1024. I'm wondering if there's a mathematical explanation for the relationship or if it's just coincidence.
Max Heinritz
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If any integer to the power of $x$ is integer, must $x$ be integer?

My apologies if this has been asked already, I've searched but couldn't find it... Let $x$ such that for every $y \in N$, $y^x$ is an integer. Does that necessarily mean that $x$ is an integer?
dkb
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Prove exponent law $a^b\cdot a^c=a^{b+c}$ for all $ a, b \text{ and } c \in \mathbb{R}$ and $a>1$

For all $ a, b \text{ and } c \in \mathbb{R}$ and $a>1$, Prove that $a^b\cdot a^c=a^{b+c}$ I have come across this question and its bugging me. Its a basic property that we learn in HS and I was hoping someone can enlighten me
math101
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"Wild" exponents of $e$

One of the things I'm curious about is why do some functions describe something like this: $$f(x,y) = e^{-\frac{x^2+y^2}{2\sigma^2}}$$ And people mostly take it for granted, throw it around for various kernels for processing images... But nobody…
ElcorAllFour
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What is the shortest way to compute the last 3 digits of $17^{256}$?

What is the shortest way to compute the last 3 digits of $17^{256}$ ? My solution: \begin{align} 17^{256} &=289^{128} \\ &=(290 - 1)^{128}\\ &=\binom{128}{0}290^{128} - ... +\binom{128}{126}290^2 - \binom{128}{127}290 +…
user2369284
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What's the first digit of 2410^2410?

The first digit means the left most digit. 2410 is just an example and it can be replaced by any other numbers. Can any one help me to solve it?
daizuozhuo
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Power term calculated from preceding terms

My 11 year old son was playing around with powers of 3 (he's like that) and came up with an interesting pattern. We worked together to extend it and came up with this observation: $$a^b = 1 + (a-1) \sum_{n=0}^{b-1} a^n$$ where 'a' and 'b' are…
Swanny
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Why does any nonzero number to the zeroth power = 1?

I can't properly wrap my head around this odd concept, though I feel I'm almost there. A non-zero base raised to the power of 0 always results in 1. After some thinking, I figured this is the proof: $\frac{{x}^{2}}{{x}^{2}}=…
J. Doerty
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