Simplify root(3)(48b^10)
Find factors of 48 that are perfect cubes
(the cube of 2 is 2^3=8, the cube of 3 is 3^3=27, etc).
48 = 8 *6 = 2^3 *6
The cube root of 2^3 is 2. The2^3 comes "out" of the cube root symbol as a 2, while the 6 stays inside.
root(3)(48b^10)=root(3)(8*6b^10)=root(3)(color(red)(2^3)*6b^10)=color(red)2root(3)(6b^10)
Next lets look at the cube root of b^10.
If you divide 10 by 3, the result is 3 with a remainder of 1.
In other words b^10=b^3*b^3*b^3*b^1. Each b^3 is a perfect cube, and comes "out" of the cube root symbol as b.
2root(3)(6b^10)=2root(3)(6color(red)(b^3)*color(blue)(b^3)*color(magenta)(b^3)*b^1)=2color(red)b*color(blue)b*color(magenta)broot(3)(6b)
Multiplying the three b's outside the cube root symbol gives
2b^3root(3)(6b)