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I came across this fraction after practising with bunch for a while. How do I write this fraction $$1\over x^a$$ as $$x^n$$ What happens to the $a$? I'm confused.

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4 Answers 4

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By definition: $$\frac1{x^a}=x^{-a}$$ So $n=-a$. The $x^{-a}$ form is usually used when there arises the need to save space.

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$$\frac{1}{x^a}=\frac{x^0}{x^a}=x^{0-a}=x^{-a}$$

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As $\frac {1}{x}=x^{-1}$, we can see that $\frac {1}{x^a}=(x^{-1})^a$, which can be simplified to $x^{-a}$. Note; $1^a = 1, (x^m)^n=x^{mn}$

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You want to find $n$ such that $$x^n=\frac{1}{x^a}$$ $$x^nx^a=1$$ $$x^{n+a}=1$$ So, you have to take $n$ such that $n+a=0$, i.e. $n=-a$.

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