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I found this identity while reading a proof.

The context: $-1\leq x<0$. I am unable to see this identity being true intuitively. Can someone show me a proof of this, or how to get from the left side of the equality to the right side?

janmarqz
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John Mayne
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  • $(-1)^n-x^{n+1} = (-1)^n(-1)x^{n+1} = (-1)^{x+1}x^{n+1} = [(-1)*x]^{n+1} = (-x)^{n+1}$.. Thing to realize is that $-x^{n+1}$ means $-(x^{n+1})$ and does not mean $(-x)^{n+1}$. – fleablood Sep 07 '16 at 16:52

3 Answers3

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$$(-1)^n \cdot -x^{n+1} = (-1)^n \cdot (-1)\cdot x^{n+1} = (-1)^{n+1} x^{n+1} = (-x)^{n+1}$$

Note that $-x^{n+1}$ is not the same as $(-x)^{n+1}$.

Alexis Olson
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$$ (-1)^{n}\cdot (-x^{n+1})=(-1)^{n}\cdot((-1)x^{n+1})=((-1)^{n}(-1))\cdot x^{n+1}=(-1)^{n+1}x^{n+1}=((-1)x)^{n+1}=(-x)^{n+1} $$

ervx
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we have $$(-1)\cdot (-1)^n\cdot x^{n+1}=(-1)^{n+1}x^{n+1}=(-x)^{n+1}$$

Dr. Sonnhard Graubner
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