Simplifying exponential fraction

Question

I have this exponential fraction $$\frac{2^{n+1}}{5^{n-1}}$$

I was wondering how we simplify something like this.

I know if the top and bottom had the same like $\frac{2^{n+1}}{2^{n+1}}$, you would just subtract the exponent.

But in my situation, I'm not too sure how to tackle it.

Answer 1

$\frac{2^{n+1}}{5^{n-1}}=2\times 5\times \frac{2^n}{5^n}=10\times \left(\frac{2}{5}\right)^n=10\times 0.4^n$ if you wish. But if you are dealing with simplifying fractions, I do think your answer is fine.

Answer 2

You cannot. The most ''simple'' forms you can achieve are

$$ 2^2\frac{2^{n-1}}{5^{n-1}}=4\left(\frac{2}{5} \right)^{n-1} $$

and doing something similar

$$ 25 \left(\frac{2}{5}\right)^{n+1}. $$

Answer 3

If the powers of the numerator and denominator matched, then you could combine them into a single fraction to that power.

$$ \frac{a^n}{b^n} = \left( \frac{a}{b} \right)^n $$

In your case we need to manipulate the fraction a bit first since the powers don't match.

I will show you how to do this with a similar example. You can then try it on your own problem.

$$ \frac{3^n}{4^{n-2}} = \frac{3^n}{4^n 4^{-2}} = \frac{1}{4^{-2}} \frac{3^n}{4^n} = \frac{4^2}{1} \frac{3^n}{4^n} = 16 \left(\frac{3}{4}\right)^n$$