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We know that powers are the number of times a number is multiplied with itself. Like $2^3$ means $(2×2×2)$, $5^4$ means $(5×5×5×5)$, etc. But how do negative powers make sense? What does it mean to multiply a number negative number of times by itself? And how are they the reciprocal of their positive counterparts?

Same question for fractional powers, how come they end up as roots?

AltercatingCurrent
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    For every real $n$ , we have $a^{-n}=\frac{1}{a^n}$ if $a>0$. This follows from the deifinition of $a^0=1$ and the power rule $a^m\cdot a^n=a^{m+n}$ giving $a^{-n}\cdot a^n=a^0=1$ – Peter Dec 08 '21 at 14:53
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    $2^{-3}=(2^3)^{-1}$ is the number who when multiplied by $2^3$ will result in $(2^3)^{-1}\times 2^3 = 1$. You can think of it as $2^{-3} = \frac{1}{2}\times\frac{1}{2}\times\frac{1}{2}$ – JMoravitz Dec 08 '21 at 14:54
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    $a^{1/n}$ is the same as the $n$-th root of $a$ because of $(a^{1/n})^n=a^{1/n\cdot n}=a^1=a$ – Peter Dec 08 '21 at 14:56
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    "What does it mean to multiply a number negative number of times by itself" The correlation between exponentiation and "multiplying a number by itself" only really works as an analogy for case of natural numbers as exponents. You should discard that analogy and grow to use the actual definition when moving on to other cases. – JMoravitz Dec 08 '21 at 14:57
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    Here is another explanation. – ryang Dec 08 '21 at 15:43

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