[Math] Simplifying a ratio of powers


This might sound like a stupid question but when it comes to simplifying when using the ratio test I get confused. Can someone please explain why $$\frac{2^{n+1}}{2^n}=\frac{2}{1}?$$ I think I might be thinking too hard because this confuses me.

Best Answer

It's the rule for dividing exponents. $\frac{x^a}{x^b}=x^{a-b}$.

You can also think of it this way:

$$\frac{2^{n+1}}{2^n}=\frac{2\cdot \cdots \cdot 2 \cdot 2}{2\cdot \cdots \cdot 2}$$

where the number $2$ occurs $n+1$ times in the numerator and $n$ times in the denominator. Exactly $n$ of those cancel, leaving a single $2$ on top, and nothing but $1$ on the bottom.

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