For positive integers $n,k$ with $k\ge3$ prove $k^{n+k}>(n+k)^k$

Question

For positive integers $n,k$ with $k\ge3$ prove by induction that $k^{n+k}>(n+k)^k$.

What is unclear to me is the induction step.

My attempt: Taking $n+k=m$ and k=3 we are going to prove the base step by induction, i.e that for $m\ge4$ we have $3^m>m^3$ The claim is true for m=4. We are going to prove that if it's true for $m$ then it will be true for $m+1$. Let's assume it is. We have $(m+1)^3=m^3+3m^2+3m+1<3^m+m^3+0.5m^3+0.5m^3<3*3^m=3^{m+1}$

But now I don't know what to do

Answer 1

Hint.

From

$$ k^n k^k > (n+k)^k\to k^n > \left(\frac nk+1\right)^k $$

and

$$ \left(\frac nk+1\right)^k < e^n $$

Answer 2

Hint: The function $f(x)=x^{1/x}$ is decreasing for $x \ge e$.