limit of a sequence by the nth root rule

Question

this question comes after the section in limits concerning limit laws, cesaro limit, and the sandwich rule. $$\displaystyle{\lim _{n\to \infty }\sqrt[n]{4^n+7^n}}$$ the first thing i tried was by the rule $\displaystyle{\lim_{n\to\infty}\sqrt[n]{a_n}=\lim_{n\to\infty}\frac{a_n}{a_{n-1}}}$ . so i get $$=\lim _{n\to \infty }\frac{4^n+7^n}{4^{n-1}+7^{n-1}}$$ but from here im not sure what to do, so i took $4^n$ out of the root and got $\sqrt[n]{4^n(1+(\frac{7}{4})^n)}=4\sqrt[n]{1+(\frac{7}{4})^n}$ . but even now im not sure what to do?

Answer 1

The first attempt can work—hint: divide the numerator and denominator by $7^n$.

The second approach can also be made to work, though I suggest factoring $7^n$ out of the root instead of $4^n$.