$\displaystyle \sqrt[4]{16a^7b^8}$$\displaystyle =$$\displaystyle 2ab^2 \sqrt[4]{a^3}
$
I don't know how to make the jump from the first thing to the last thing.
EDIT: There was a typo, sorry.
For sure!
Well because $\displaystyle a^3a^4=a^7$ Just a small expansion.
$\displaystyle
\sqrt[4]{16a^3a^4b^8}
$
By the rule $\displaystyle \sqrt{ab}=\sqrt{a}\sqrt{b}$, so we can split up the expression and take the fourth root of each term.
For additional clarification, I will rewrite that.
$\displaystyle
\sqrt[4]{16}\sqrt[4]{a^3}\sqrt[4]{a^4}\sqrt[4]{b^8}
$$\displaystyle
=\sqrt[4]{2^4}\sqrt[4]{a^3}\sqrt[4]{a^4}\sqrt[4]{b^8}
$$\displaystyle =2ab^2\sqrt[4]{a^3}$
Does this help, or should I clarify more?