Simplifying Expression: $(-1)^n/(4^n)n(-4)^n$ - Explained

[tmt1]

I have

$$\frac{(-1)^n}{(4^n)n }(-4)^n$$ (where n is a sufficiently large positive number, I think in this case it only has to be positive).

Is $\frac{-4^n}{4^n}$ the same thing as $(-1)^n$?

How is this the case?

 

[soroban]

I have: $\frac{(-1)^n}{(4^n)n }(-4)^n$ (where n is a sufficiently large positive number,
I think in this case it only has to be positive).

Is $\frac{-4^n}{4^n}$ the same thing as $(-1)^n$?

How is this the case?

$$\text{Recall this rule: }\;\frac{a^n}{b^n} \;=\;\left(\frac{a}{b}\right)^n$$

$$\text{We have: }\;\frac{(-4)^n}{4^n} \;=\; \left(\frac{-4}{4}\right)^n \;=\;(-1)^n$$

 

[Greg]

Be careful with the parentheses! They make a big difference.

$$\dfrac{(-4)^n}{4^n}=\left(\dfrac{-4}{4}\right)^n=(-1)^n$$

$$\dfrac{-4^n}{4^n}=-\dfrac{4^n}{4^n}=-\left(\dfrac44\right)^n=-1$$

Simply put, $(-1)^n$ equals 1 when $n$ is even and -1 when $n$ is odd. $-1^n=-1$

Okay?

^Looks like soroban got there first!