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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.

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3 Answers 3

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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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Here, we have \begin{align} \frac{2^{n+1}}{2^n}&=2^{(n+1)-n} & \text{using exponent law $x^{a-b}=\frac{x^a}{x^b}$}\\ &=2^1 & \\ &=\frac{2}{1} & \text{because } 2^1=2=\frac{2}{1} \end{align}

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Think about it this way. When you write $2^{n+1}$ you are multiplying a bunch of $2$s in sequence. Same for $2^n$. So if we write that as a bunch of $2$s:

(for $n=5$)

$\frac{2 * 2 * 2 * 2 * 2 * 2}{2 * 2 * 2 * 2 * 2}$

We can cancel out all of the 2s on the bottom and leave a single $2$ on top. And that single $2$ is the same as $2/1$.

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