Is there a way to simplify this exponent?

Question

I have the exponents $k^4+k^5+k^6$. Is there any way to simplify this into one exponent? I'm trying to find a way to simplify a sequence of increasing exponents into one, but am not sure how. With some guess and checking, I got $k^4+k^5+k^6 \approx k^{6.13368}$, but why does this sequence simplify to approximately $k^{6.13368}$, and how can I simplify other sequences, like $k^6+k^7+k^8+k^9$?

Answer 1

There's no way to simplify the expression $P(k)=k^2+k^4+k^6$. However, you can ask a related question: given some particular $k=k_0$, find the exponent $a$ such that

$$ (k_0)^a=P(k_0) $$

Taking the logarithm of both sides and solving for $a$ gives

$$ a=\frac{\ln{P(k_0)}}{\ln{k_0}} $$

Substituting $k_0=5$ into the above expression, I find $a=6.133656\dots$, which is in agreement with the exponent you found from trial-end-error.