If an 8-foot ladder is leaned against a wall with its foot braced 4 feet from the wall's base, how far up the wall will the top of the ladder reach? round to the nearest hundredth.

We know that
see the picture to better understand the problem
in the right triangle ABC
Applying the Pythagoras Theorem
AB²=BC²+AC²-------> BC²=AB²-AC²
AB=8 ft
AC=4 ft
BC²=8²-4²-----> BC²=48--------> BC=√48-------> BC=6.93 ft

the answer is
6.93 ft